V.I.Romanovskiy Institute of Mathematics, Uzbek Academy of Sciences
National University of Uzbekistan named after Mirzo Ulugbek
Samarkand State University named after Sh.R.Rashidov

ABSTRACTS
OF THE INTERNATIONAL CONFERENCEBRANCHING PROCESSESAND THEIR APPLICATIONS

September 18-22, 2023
Tashkent-Samarkand, Uzbekistan

INTERNATIONAL CONFERENCE"BRANCHING PROCESSES AND THEIR APPLICATIONS"

Organizing Committee:
Institute of Mathematics of the Academy of Sciences of Uzbekistan:Ayupov Sh.A. – Chairman
Khusanbaev Ya.M. – Vice-Chairman
Formanov Sh.K.
Rozikov U.A.
Ganikhodjaev N.N.
National University of Uzbekistan:
Madjidov I.U.
Ergashov Y.S.
Zikirov O.S.
Sharipov O.Sh.
Samarkand State University:
Khalmuradov R.I.
Khushvaktov H.
Soleev A.S.
Mo’minov M.E.
Kuljonov U.N.
Qudratov H.E.

International Sceintific Committee:
Jean Bertoin – Committee Chair,(Institute of Mathematics, Universitety Zurich, Switzerland )Vitali Wachtel (Bielefeld University, Bielefeld, Germany)
Vincent Bansaye (Ecole Polytechnique, Palaiseau, France)
Zenghu Li (Beijing Normal University, Beijing, China)
Marek Kimmel (Rice University, Houston, USA)
Vladimir Vatutin (Steklov Mathematical Institute, Moscow, Russia)
Serik Sagitov (Chalmers University, Gothenburg, Sweden)
Alexey Lindo (School of Mathematics and Statistics, Glasgow, UK )Ibrahim Rahimov (Institute of Mathematics, Tashkent, Uzbekistan)

The authors are responsible for the accuracy of the information provided in
the conference proceedings.

Contents
1 Branching processes
7
Afanasyev V.
Weakly supercritical branching process in non-favorable random
environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Aliyev S.
Limit theorem for a subcritical branching process with continuous
time and migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Azimov J.
Limit theorems for the branching process with decreasing
immigration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Bauman N., Chigansky P., Klebaner F.
Approximation of size-density dependent branching processes . . 14
Braunsteins P., Hautphenne S., Minuesa C.
Consistent estimation for population-size-dependent branchingprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Bulinskaya E.
Asymptotic Shape of Branching Random Walks on Periodic Graphs 17
Cloez B.
Recent results on eigenvalues for branching processes and related
fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Denisov K.
Lower Large Deviations of Strongly Supercritical Branching
Process in Random Environment with Geometric Number ofDescendants: Local Asymptotics . . . . . . . . . . . . . . . . . . . . . 20
Formanov Sh., Khusanbaev Y.
On reduced processes starting from a large number of particles . 22
Foucart C.
Explosions and dualities in logistic continuous state branchingprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
González M., Minuesa C., del Puerto I., Vidyashnakar A.
Large Deviation for Supercritical Controlled Branching Processes 24
Wenming Hong
Conditional central limit theorem for critical and subcriticalbranching random walk . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Imomov A.
On explicit expression of the Generating Function of Invariant
Measures of Critical Galton-Watson Branching Systems . . . . . . 26
3

4

CONTENTS

Imomov A., Murtazaev M.
On refinement of some limit theorems for the noncritical GaltonWatson Branching Systems . . . . . . . . . . . . . . . . . . . . . . . . 27
Ispány M., Bondon P., Reisen V.
Periodic branching processes with immigration and their implicit
multi-type representation . . . . . . . . . . . . . . . . . . . . . . . . . 29
Juraev Sh.
On refinements of the asymptotic expansion of the continuationof critical branching processes . . . . . . . . . . . . . . . . . . . . . . . 31
Kolb M.
Explosion phenomena for Fleming-Viot-type processes . . . . . . . 33
Kudratov Kh., Khusanbaev Y.
A limit theorem for the critical Galton-Watson branching processes 34
Lukashova I.
A periodic branching random walk with immigration on Zd . . . . 36
Meiners M.
Asymptotic fluctuations of supercritical general branchingprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Méléard S.
Time reversal of spinal processes for linear and non-linear
branching processes near stationarity . . . . . . . . . . . . . . . . . . 37
Minuesa C., González M., del Puerto I.
Bayesian inference in controlled branching processes via ABC
methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Nazarov Z.
The total progeny in the positive recurrent Q-processes . . . . . . 38
Penington S.
Gaussian waves in BBM with mean-dependent branching . . . . . 41
Rahimov I., Sharipov S.
Functional limit theorems and the asymptotic normality of
estimators based on partial observations . . . . . . . . . . . . . . . . 41
Rasulov A., Raimova G.
Branching process for the solution of semi-linear Helmholtz
boundary value problem . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Sakhanenko A.
On random walks in random environment with random local
constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Schertzer E.
Pushed and pulled waves in population genetics . . . . . . . . . . . 46
Schweinsberg J., Shuai Y.
Asymptotics for the site frequency spectrum associated with thegenealogy of a birth and death process . . . . . . . . . . . . . . . . . 47
Shklyaev A.
Large Deviations of Bisexual Brancing Processes in Random
Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Sumit Kumar Yadav
Coalescence in Bisexual Branching Processes . . . . . . . . . . . . . 49

CONTENTS

Toshkulov Kh., Khusanbaev Y.
On the rate of convergence in limit theorems for fluctuation
critical branching processes with immigration . . . . . . . . . . . . .
Vatutin V., Dyakonova E., Dong C.
Random walks conditioned to stay nonnegative and branchingprocesses in nonfavorable random environment . . . . . . . . . . . .
Wang Hua-Ming
Times of a branching process with immigration in varying
environment attaining a fixed level . . . . . . . . . . . . . . . . . . .
Wang Ren-Yi, Kimmel Marek
A Countable-Type Branching Process Model for the Tug-of-War
Cancer Cell Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wachtel V.
Near-critical branching processes considered as Markov chainswith small drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Watson A.
Growth-fragmentation and quasi-stationary methods . . . . . . . .
Yarovaya E.
Spectral methods and their applications in the theory of
branching random walks . . . . . . . . . . . . . . . . . . . . . . . . . .
Yanev G.
Branching Processes with Migration Subordinated by Renewal
Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Zhumayev Y.
Theta positive branching processes in varying environment . . . .
Zuyev S., Molchanov I.
Branching selfdecomposability and limit theorems forsuperposition of point processes . . . . . . . . . . . . . . . . . . . . .
2 Stochastic analysis
Adirov T.
Estimation of the Parameter of one Class of Distributions . . . . .
Aliyev Y.
Digits of Powers of 2 in Ternary Numeral System . . . . . . . . . .
Aliyev A., Tirkasheva G.
Central limit theorem for stochastic perturbations of PL circle
maps with two break points . . . . . . . . . . . . . . . . . . . . . . . .
Delmas Jean-François
SIS model on (random) dense graphs: probabilistic approach andremarks on optimal vaccinations . . . . . . . . . . . . . . . . . . . . .
Dzhalilov A., Abdukhakimov S.
Note on critical mappings of the circle with rotation number ofalgebraic type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gafurov M. U.
Convergence rate estimates in the Hartman-Wintner Law of the
Iterated Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ganikhodjaev N.
Quadratic Stochastic Operators with Countable State Space . . .

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50

51

52

53

53
54

54

55
56

58
61
62
63

65

66

67

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69

6

CONTENTS

Horton E.
Long-term behaviour of the neutron transport equation at
criticality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Khamdamov I.
Weakly Dependent Properties of Vertex Processes of a Convex
Hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Khodjibayev V., Lotov V.
On the distribution of the crossing number of a strip bytrajectories of the Levy process . . . . . . . . . . . . . . . . . . . . . .
Korablina Y.
Boundedness of the classical operators in weighted quasi-Banach
spaces of entire functions . . . . . . . . . . . . . . . . . . . . . . . . . .
Litvinov V., Litvinova K.
Stochastic longitudinal oscillations viscoelastic rope with moving
boundaries, taking into account damping forces . . . . . . . . . . .
Mamatov Kh.
Limited Theorem for Stochastic Integrals over Semi-Martingales .
Mamurov I.
Limited Theorem for Members of the Variational Series with aRandom Sample Size . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mirakhmedov Sh.
Asymptotic approximation associated with generalized random
allocation schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mizomov I.
On the identity of the theta functions . . . . . . . . . . . . . . . . . .
Rozikov U., Olimov U.
Dynamical system of an infinite-dimensional operator in an
invariant set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sattarov A.
Classification of non-strongly nilpotent filiform Leibniz algebras
of dimension 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sharakhmetov Sh.
Extreme values of functionals of integral form . . . . . . . . . . . .
Sharipov O., Kobilov U.
On the maximum of dependent random variables . . . . . . . . . .
Sharipov O.Sh., Muxtorov I.G‘.
Central limit theorem for ρ-mixing random variables with values
in Lp [0, 1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Zuparov T.
Central limit theorem for a 1-order autoregressive process withrandom coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Path 1

Branching processes

7

Weakly supercritical branching process in
non-favorable random environmentAfanasyev, V. I.
Steklov Mathematical Institute of Russian Academy of Sciences
viafan@mail.ru

Let (Ω, F, P) be a probability space and ∆ be the space of probability measureson N0 = {0, 1, 2, . . .} equipped with the metric of total variation. Consider the random
elements Q1 , Q2 , . . ., mapping (Ω, F, P) into ∆. The sequence Π = {Q1 , Q2 , . . .} is calleda random environment.
A sequence of non-negative integer random variables {Zn , n ∈ N0 } is called a branchingprocess in random environment (BPRE) if Z0 = 1 andZn
X(n)
Zn+1 =
ξi ,n ∈ N0 ,

i=1

where
it is assumed that,no conditioned on the random environment Π, the random variables(n)(n) (n)ξi , i ∈ N, n ∈ N0 are independent and for a fixed n ∈ N0 variables ξ1 , ξ2 , . . . are
identically distributed with distribution Qn+1 .
In the language of branching processes, Zn is the number of particles of the nth(n)generation, ξi is the number of direct descendants of the ith particle from the nthgeneration. We will study the described model under the assumption that the random
elements Q1 , Q2 , . . . are independent and equally distributed.
Set for i ∈ N
ϕ00 (1)
Xi = ln ϕ0i (1) , ηi = 0i(ϕi (1))2(we suppose that ϕ01 (1) , ϕ001 (1) ∈+∞) a.s.). We introduce the so-called associatedP(0,nrandom walk : S0 = 0, Sn =i=1 Xi for n ∈ N. Note that the random vectors(X1 , η1 ) , (X2 , η2 ) , . . . are independent and identically distributed for the considered
BPRE. We indicate the assumptions used in the paper regarding the random vector(X1 , η1 ).
Assumption A. The process {Zi , i ∈ N0 } is weakly supercritical, i.e. EX1 > 0 andEX1 e−βX1 = 0 for some β ∈ (0, 1).
Set γ = Ee−βX1 . Let F (·) be the distribution function of a random variable X1 . We
introduce the distribution function
F (β) (x) = γ −1Zxe−βu dF (u) , x ∈ R.

−∞

Assumption B. The distribution F (β) (x) belongs to the domain of attraction of somestable law with the index α ∈ (1, 2] and is non-lattice.
8

Assumption C. For some ε > 0hi
α+εE ln+ η1
exp (−βX1 ) < +∞.

We introduce a random process Yn :Yn (t) =Zbntc
 , t ∈ [0, 1) ;exp Sbntc

Yn (1) = Zn .

Let the symbol ⇒ denote convergence in the sense of finite-dimensional distributions. We
formulate the main results. Set for n ∈ N
Mn = max Si .
1≤i≤n

The associated random walk {Sn } of the supercritical BPRE has a positive drift andtherefore Sn and Mn tend to +∞, as n → ∞. In this paper, a weakly supercriticalbranching process is considered in non-favorable random environment of two types, wheneither Mn < 0 or Sn ≤ u, where u is a positive constant. For the random process{Yn (t) , t ∈ [0, 1]}, the following functional limit theorems are proved in the case of nonfavorable environments of these types.
Theorem 1. If assumptions A, B and C are valid, then for any u ∈ [−∞, 0), as
n → ∞,{Y (t) , t ∈ [0, 1] | Mn < 0, Sn ≥ u} ⇒ {Uu (t) , t ∈ [0, 1]} ,where {Uu (t) , t ∈ [0, 1]} is a random process with non-negative constant trajectories on
the interval (0, 1), and the probability of the event {Uu (t) > 0} is positive for t ∈ (0, 1); therandom variable Uu (1) takes values from N0 and the probability of the event {Uu (1) > 0}is positive.
Set
Tn = max {i : Si = max (0, Mn ) , 0 ≤ i ≤ n} ,that is, Tn is the last moment when the maximum of the random walk S0 , . . . , Sn is
attained.
Theorem 2. If assumptions A, B and C are valid, then for any u ∈ (0, +∞), as
n → ∞,{Yn (t) , t ∈ [0, 1] | Tn = n, Sn ≤ u} ⇒ {Vu (t) , t ∈ [0, 1]} ,where {Vu (t) , t ∈ [0, 1]} is a random process with non-negative constant trajectories on
the interval (0, 1), and the probability of the event {Vu (t) > 0} is positive for t ∈ (0, 1); therandom variable Vu (1) takes values from N0 and the probability of the event {Vu (1) > 0}is positive.
This work is supported by the Russian Science Foundation under grant no. 19-1100111-II, https://rscf.ru/project/19-11-00111/.

9

Limit theorem for a subcritical branching process with
continuous time and migration
Soltan A. Aliyev
Institute of Mathematics and Mechanics, Baku, Azerbaijan
Institute of Control Systems Baku, Azerbaijansoltanaliyev@yahoo.com

Keywords: branching process, limit theorem, migration, subcritical.
Branching process are mathematical models of many physical, biological, genetic,
demographic and other processes. Since third- party factors often exist, there is a need to
study different modifications of this process. Among them are branching processes, withimmigration, emigration, or a combination of processes, namely processes with migration
for the case of discrete and conditions time.
An important feature of the branching process is the generating function. In theclassical case, for processes with continuous time, it is obtained from the differential
equation.
In the case of the branching processes with immigration, the derivation of thedifferential equation and finding its solution in [1], where the process is defined as aprocess with two types of particles.
In the case of the process of emigration, Formanov Sh.K. and Kaverin S.V. found theform of a differential equation and the solution of this equation without detained inferenceis shown in [3],[4].
The main results for branching processes with discrete time and different regimes ofimmigration and emigration are described in [5].
The limit distribution theorem for the classical branching process with continuous timeis proved in [2]. In this work we consider a more general model of the branching processes
with migration and continuous time [6].
Immigration, emigration and evolution occur at random moments of time and aredetermined by the intensity of the transition probabilities.
The form of a generating function for a branching process with migration andcontinuous time and the Kolmagorov system of equation held for the transition
probabilities of the process are found in [6].
Consider a Markov branching process with one type of particles and migration µ(t), t ∈[0, ∞). Let µ(t) denote the number of particles at the time t ∈ [0, ∞).
We suppose, that at the time t = 0, the process starts with one particle in the systemµ(0) = 1.
The process µ(t), t ∈ [0, ∞) then ∆t → 0 is given by transition probabilities
P {µ(t + ∆t) = j/µ(t) = i} = Pij (t), which are expressed by intensity of reproduction
particle pk (k = 0, 1, ...), the intensity of immigration qk (k = 0, 1, ...), and the intensity ofemigration rn (n = 0, m) [7].
pk , qk , rn satisfy the conditions
pk ≥ 0, k 6= 1, p1 < 0,

∞
X
k=0

10

pk = 0,qk ≥ 0, k 6= 0, q0 < 0,∞X

qk = 0,k=0

rn ≥ 0, n = 1, m, r0 < 0,m
X

rk = 0,k=0

We introduce the following notationf (s) =∞
X

pn sn , |s| ≤ 1,n=0g(s) =∞
X

qn sn , |s| ≤ 1,n=0r(s) =mX

qn s−n , 0 ≤ |s| ≤ 1,

n=0

We consider the subcritical branching process and find limiting distribution
R∞
Theorem. If a0 = f 0 (1) < 0, a1 = g 0 (1) < ∞, a2 = r0 (1) < ∞ and M µ(x)dx < ∞,0then limiting distribution µ(t) exists
lim P {µ(t) = n} = Pn∗ .

t→∞References:
1. Sevastyanov, B.A.: Branching processes, Izdat. Nauka, Moscow, 436 pp. (1971) (InRussian)
2. Sevastyanov, B. A.: Limit theorems for branching stochastic processes of special form,
Theory Probab. Appl., 2(3), 339-348 (1957). (In Russian)
3. Formanov, Sh.K., Kaverin S. V.: Markov branching processes with emigration. I,
Izvestiya of the Academy of Sciences of the USSR. Avg. Physical motif. Sciences, 5, 23-28(1986). (In Russian)
4. Formanov, Sh.K., Kaverin, S. V.: Markov branching processes with emigration. II,
Izvestiya of the Academy of Sciences of the USSR. Avg. Physical motif. Sciences, 3, 36-41(1987). (In Russian)
5. Yanev, G.P.: Critical controlled branching processes and their relatives, Pliska Studia
Mathematica., 24, 111-130 (2015).
6. Yakymyshyn, K.: Equation for generation function for branching processes with
migration, Visnyk of the Lviv Univ. Series Mech. Math., 84, 119-125 (2017). (in Ukraine)
7. Khrystyna M. Yakymyshyn, Iryna B. Bazylevych, Soltan A. Aliyev, Limit theorems
for homogeneous branching processes with migration, Trans. Natl. Acad. Sci. Azerb. Ser.
Phys.-Tech. Math. Sci. Mathematics, 41 (4), 141-152 (2021)

11

Limit theorems for the branching process withdecreasing immigration
Azimov Jakhongir
Tashkent State Transport University
azimovjb@gmail.com

Keywords: Branching random process, state-dependent immigration, slowly varying
function, generating function.
Let µn be a number of particles of the Galton-Watson (G-W) branching process at the
moment n (n = 0, 1, . . . , µ0 = 1 ) with the generating function (g.f.)
F (x) =∞
X

p j xj ,pj = P {µ1 = j} ,j = 0, 1, ...,

|x| ≤ 1.

j=0

The zero state is absorbing for the process µn , that is, if µN = 0 for some N > 0,then µn = 0 for all n > N. In [1] J.H.Foster considered G-W process modified to allow
immigration of particles whenever the number of particles is zero. If µn = 0, then, at
the moment n, ξn particles immigrate to the population, where the number of particles
evolves by the law of the G-W process with g.f. F (x ).
The asymptotic behavior of branching processes with state-dependent immigration
were studied by many authors (see [1]–[3]).
We consider the case when immigration takes place as µ n =k, 0 ≤ k ≤ m, where m issome nonnegative integer. Assume that the intensity of the immigration decreases tending
to 0, when the number of descendents increases. Limit theorems for such processes havebeen studied in [4],[5],[6].
Thus, the immigration is given with g.f.
gk,n (x) =∞X

qkj (n)xj ,|x| ≤ 1,

k = 0, 1, ..., m,qkj (n) ≥ 0,

j=0
∞
X

qkj (n) = 1,n = 0, 1, 2, ...

.

j=0

Let{Zn ; n = 0, 1, ...} be a number of particles of this process at the moment n.
Suppose, that
F (x) = x + (1 − x)1+ν L(1 − x)
where 0 < ν ≤ 1 and L(x) is a slowly varying function (s.v.f.) as x → 0.
Introduce the function
n
X
N (k)
M (n) =,k 1/νk=1
where N (x) is a s.v.f. as x → ∞ such that νN ν (x)L(N (x)/x1/ν ) → 1.
Denote000
αn = max gk,n (1)βn = max gk,n (1),
0≤k≤m

0≤k≤n

12

Q1 (n) = αnnX

(1 − Fk (0)),

Q2 (n) = (1 − Fn (0))

k=0where F0 (x) = x,
We suppose that

n
X

αk

k=0

Fn+1 (x) = F (Fn (x)).
Sup αn < ∞ ,Sup βn < ∞ ,nn

0 < αn → 0 ,βn → 0 ,We consider the case ν = 1,
Theorem 1. Letn → ∞.

M (n) → ∞ as n → ∞.

l(n), βn = o (Q1 (n)) , n → ∞,nr
where 0≤r <1 and l(n) is a s.v.f. as n → ∞. Then for all 0≤х ≤1
M (Zn )
< x/Zn > 0 = x.
lim P
n→∞
M (n)
αn ∼

Theorem 2. Let αn ∼ Nn(n) , βn = o (Q1 (n)) ,Q1 (n)
n → ∞ and lim Q
= a,
2 (n)
n→∞

a ≥ 0.

Thenа) for all 0<х<1,

lim P

n→∞


ax
M (Zn )
< x/Zn > 0 =
;
M (n)
1+a

б) for х ≥ 0,

lim P

n→∞


N (n)Zna1 − e−x< x/Zn > 0 =
.
+
n
1+a
1+a
References:

1. Foster J. A limit theorem for a branching process with state-dependent immigration,
Ann.Math.Stat.,, Vol.42, 1971, year, pp. 1773–1776.
2.Pakes A. A branching processes with state-dependent immigration component,
Adv.Appl.Prob., No 3., 1971, year, pp. 301–314.
3. Formanov Sh.K., Azimov J.B. Markov branching processes with a regularly varying
generating function and immigration of a special form,Theor.Prob. and Math.Stat.,, Vol.
65, 2002, pp. 181–188.
4. Mitov K.,Yanev N., Critical Galton-Watson processes with decreasing state-dependentimmigrations. J.Appl.Prob., 21,1984,22-39. pp. 181–188.
5. Mitov K., Vatutin V., Yanev N. Continuous time branching processes with decreasingstate-dependent immigrations. Adv.Appl.Prob.,, 16, 1984. pp. 697–714.
6. Azimov J. Asymptotic properties of the branching processes with state–dependentimmigrations. Proceedings of the Fourth International Conference “Problems of
Cybernetics and Informatics”,, September 12-14, 2012, Baku, Azerbaijan. pp. 161164.

13

Approximation of size-density dependent branchingprocesses
Bauman, Naor; Chigansky, Pavel; Klebaner, Fima
The Hebrew University of Jerusalem, Israel
The Hebrew University of Jerusalem, Israel
Monash University, Australia
Naor.Bauman@mail.huji.ac.il, Pavel.Chigansky@mail.huji.ac.il, Fima.Klebaner@monash.edu

Keywords: size-density dependent branching processes, limit theorems
Consider the branching process (Zn ) generated by the recursion
Zn−1

Zn =

X

ξn,j ,

n = 1, 2, ...

j=1

subject to the initial condition Z0 . The integer valued random variables ξn,j areconditionally i.i.d. given the previous generations Fn−1 = σ{ξm,j : j ∈ N, m ≤ n} andtheir common distribution depends on the size-density Z n−1 = K −1 Zn−1 with respect tothe large parameter K:
P(ξn,j = `|Fn−1 ) = p` (Z n−1 ), ` ∈ Z+ ,(1)
P∞
where Pp` (·) ≥ 0 are some functions, `=0 p` (x) = 1. If the offspring conditional meanm(x) = ∞`=0 `p` (x) is a decreasing function with ρ := m(0) > 1 and m(1) = 1, thisrandom process can be viewed as a basic model for populations on a habitat with thecarrying capacity K. Its typical trajectory started from Z0  K grows rapidly until
reaching the capacity region, where it fluctuates for a very long period of time until
eventual extinction, see e.g. [1].
This lifecycle can be described by the limit behavior of the size-density process Z nsatisfying
Zn−1

1 X
Z n = f (Z n−1 ) +ξn,j − m(Z n−1 ) , n = 1, 2, ...
K j=1
where f (x) := xm(x). The last term is of order OP (K −1/2 ) and hence, under mild
conditions,P
max Z n − xn −−−→ 0, ∀n ∈ Z+ ,
m≤n

K→∞

where the limit sequence xn solves the recursion
xn = f (xn−1 ),n = 1, 2, ...

(2)subject to x0 = limK→∞ Z 0 . The stochastic fluctuations of Z n about the deterministic√dlimit converge to a Gaussian process V = (Vn ) in distribution: K(Z n − xn ) −−−→ Vn
K→∞

where Vn satisfies the recursion, [2],Vn = f 0 (xn−1 )Vn−1 +pxn−1 σ 2 (xn−1 )Wn ,14

n = 1, 2, ...

with N (0, 1) i.i.d. random variables Wn ’s. Large Deviations analysis [3] reveals that thetime to extinction grows exponentially with K.
This theory gives an adequate picture of the population dynamics, when its initial size
is relatively large, i.e., proportional to the capacity so that x0 := limK→∞ Z 0 > 0. In thecompliment case of Z0 being small, e.g. Z0 = 1, the recursion (2) is started from x0 = 0and, consequently, the limit sequence degenerates to xn = f n (x0 ) = 0 for all n ≥ 0.
Nevertheless, when K is large but finite, some of the trajectories manage to escape theearly extinction and still reach the capacity region, albeit at a much later time, logarithmic
in K. In this talk I will present a new type of limit theorems which capture such a delayed
emergence, [4]-[6].
On a suitable probability space, (Zn ) can be coupled with an auxiliary Galton-Watsonprocess (Yn ) with the offspring mean ρ, started from Y0 = Z0 . The sequence ρ−n Yn is aconvergent martingale with the a.s. limit W = limn→∞ ρ−n Yn . Moreover, under certain
technical conditions on f , the sequence of the scaled iterates can be shown to converge toa limit:
H(x) := lim f n (x/ρn ).
n→∞Theorem 1. [6] Let n1 := n1 (K) = [logρ K] thenP−{logρ K}
Z n1 − H W ρ−−−→ 0,K→∞(3)

where {x} = x − [x] for x > 0.
In particular, when K is an integer power of ρ, this result implies that the distributionof the size-density process at times of order logρ K is close to the distribution of therandom variable H(W ). The proof of Theorem 1 is based on the long known heuristics[7], [8] according to which a size dependent population behaves initially as the GaltonWatson branching process and, if it manages to avoid extinction at this early stage, it
continues to grow towards the carrying capacity following an almost deterministic curve.
This suggests to approximate Z n by means of Y n for n ≤ nc where nc = [logρ K c ] withsome c ∈ (0, 1) and by means of f n−nc (Y nc ) for n > nc . This approximation at time n1
produces the limit of Theorem 1 if c is taken to be any number in ( 21 , 1).
This construction depends on a free parameter c and a close inspection of the proof
reveals that the best rate of convergence in (2) is OP (K −1/8 log K) and it is achieved withc = 58 . On the other hand, in the special case when Zn is a Galton-Watson process, i.e. theprobabilities in (1) are constant with resect to Z n , Heyde’s CLT [9] implies that the rateof convergence is OP (K −1/2 ). In our recent paper [10] we show how the above construction
can be modified to produce an almost optimal rate of OP (K −1/2 log K).
References:
1. K. Hamza, P. Jagers, and F. C. Klebaner, On the establishment, persistence, andinevitable extinction of populations. J. Math. Biol., 72(4):797–820, 2016.
2. F. C. Klebaner, O. Nerman. Autoregressive approximation in branching processes
with a threshold. Stochastic Process. Appl., 51(1):1–7, 1994.
3. F. C. Klebaner and O. Zeitouni. The exit problem for a class of density-dependent
branching systems. Ann. Appl. Probab., 4(4):1188–1205, 1994.
4. A. D. Barbour, P. Chigansky, and F. C. Klebaner. On the emergence of random
initial conditions in fluid limits. J. Appl. Probab., 53(4):1193–1205, 2016.
15

5. P. Chigansky, P. Jagers, and F. C. Klebaner. What can be observed in real time
PCR and when does it show? J. Math. Biol., 76(3):679–695, 2018.
6. P. Chigansky, P. Jagers, and F. C. Klebaner. Populations with interaction andenvironmental dependence: from few, (almost) independent, members into deterministic
evolution of high densities. Stoch. Models, 35(2):108–118, 2019.
7. David G. Kendall. Deterministic and stochastic epidemics in closed populations.
Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability,
1954–1955, vol. IV, pages 149–165. University of California Press, Berkeley-Los Angeles,
Calif., 1956.
8. P. Whittle. The outcome of a stochastic epidemic—a note on Bailey’s paper.
Biometrika, 42:116–122, 1955.
9. C. C. Heyde. A rate of convergence result for the super-critical Galton-Watsonprocess. J. Appl. Probability, 7:451–454, 1970.
10. N. Bauman, P. Chigansky, F. Klebaner. An approximation of populations on a
habitat with large carrying capacity. arXiv:2303.03735

Consistent estimation for population-size-dependent
branching processes
Braunsteins Peter, Hautphenne Sophie, Minuesa Carmen
University of New South Wales (Sydney)
The University of Melbourne
University of Extremadurap.braunsteins@unsw.edu.au, sophiemh@unimelb.edu.au, cminuesaa@unex.es

Keywords: Branching process; population-size-dependence; almost sure extinction;
inference; carrying capacity; Q-process.
We consider population-size-dependent branching processes (PSDBPs) which
eventually become extinct with probability one. These processes are designed to capture
the dynamics of endangered populations living in restricted habitats with a carrying
capacity. We derive maximum likelihood estimators (MLEs) for the mean number ofoffspring born to individuals when the current population size is z ≥ 1, based ona single trajectory of population size counts. Because these processes become extinct
with probability one, we show that the MLEs do not satisfy the classical consistency
property (C-consistency). This leads us to define the concept of Q-consistency, and we
prove that the MLEs are Q-consistent and asymptotically normal. Using these MLEs, wethen propose a new class of weighted least-squares C-consistent estimators for parametric
PSDBPs with logistic growth.
Our results are motivated by conservation biology, where endangered populations areoften studied because they are still alive, thereby inducing an observation bias. Throughsimulated examples, we show that our C-consistent estimators generally reduce thisbias, leading to improved estimates for important quantities such as a habitat’s carrying
capacity. We apply our methodology to estimate the carrying capacity of the Chatham
Island black robin −a population that was reduced to a single breeding female in the
1970’s, which has since recovered but is yet to reach the island’s carrying capacity
16

References:
1. Braunsteins P., Hautphenne S., Minuesa C. Parameter estimation in branchingprocesses with almost sure extinction. Bernoulli, Vol. 28, No 1, 2022, pp. 33-63.
2. Braunsteins P., Hautphenne S., Minuesa C. Consistent least squares estimation in
population-sizedependent branching processes. arXiv:2211.10898, 2022.

Asymptotic Shape ofBranching Random Walks on Periodic Graphs
Bulinskaya, Ekaterina Vl.
Lomonosov Moscow State University
bulinskaya@yandex.ru

Keywords: branching random walk, asymptotic shape, propagation front, supercriticalregime, light tails.
A multitude of publications devoted to different models of branching random walk haveappeared within the last decade (see, e.g., [1]–[5]). Any model of branching random walk(BRW) comprises two random mechanisms. The first one accounts for splitting and deathof particles and the second one for their random moving in space. Various combinationsof the two arrangements lead to different models of BRW. These models are not onlyof theoretical interest but also have numerous applications in biology, chemical kinetics,
statistical physics, homopolymers theory, queueing theory, etc. (see, e.g., [6]–[8]).
A special place in the theory of BRW is occupied by catalytic models (see, e.g.,[9] and [10]). The term “catalytic” means that there are several points of the space
where the catalysts are located and just these catalysts make a particle reaching sucha location either split or die there. At a point without a catalyst a particle may walk only.
Accordingly, the points containing catalysts are called points of catalysis or sources ofbranching and death of particles. The case of a single source of branching was investigatedin many papers (see, e.g., [11] and [12]). The description of the spread of particles
population in catalytic BRW with an arbitrary finite number of branching sources on
Z d was initiated in papers [13] and [14], and the study was accomplished in a series of
papers by the author, we refer, e.g.,to [15]–[17]. The case of an infinite set of branchingsources having periodical structure, i.e. BRW on periodic graphs, was treated for the firsttime in [18] and [19]. Those papers describe the asymptotic behavior, with respect togrowing time, of means of some functionals related to BRW on periodic graphs.
In the present talk, in contrast to [18] and [19], we consider the spread almost sure ofparticles population in BRW on periodic graphs and study the asymptotic behavior (astime t goes to infinity) of the normalized cloud of particles existing at time t. We stipulatethat the regime of branching is supercritical and the jumps of a random walk have lighttails. Under these assumptions the instant positions of all the particles at time t arenormalized by factor t−1 before letting t → ∞. The corresponding set of the normalized
particles positions at time t is denoted by Pt . We establish that
∆ (Pt , P) → 0 a.s. on event S of population non-degeneracy,17

as t → ∞,where ∆ (D, F ) is the Hausdorff distance between sets D and F belonging to Rd . The
limiting set P ⊂ Rd is called the asymptotic shape of the BRW. We prove that, in thementioned case of periodical structure, the arising set P is compact and convex. Moreover,
we also provide an explicit formula to describe it in terms of the level sets of the Perronroots of a family of some parametric quasi-nonnegative matrices. Each element of such amatrix is the Laplace transform of the intensity measure of a specified point process.
This means that the particles population spreads asymptotically linearly in time andthe shape of random cloud of particles is approximated by tP as t → ∞. Our main
result also shows that, with probability one, the cloud of normalized particles not only
becomes asymptotically close to the boundary of P but fills the whole set P. The
assumption of supercriticality guarantees the positive probability of the event S of the
population survival. On the opposite event the problem of the rate of the population
spread is meaningless because of the population degeneracy. The assumption of lighttails of the random walk jumps leads to the indicated normalizing factor. We also tackle
other conditions concerning the behavior of random walk tails (e.g., the tails are regularly
varying or the jump has a semi-exponential distribution). Such conditions, effecting the
population spread, were employed in [16] and [17] for BRW with finitely many sources ofbranching.
Note that in papers [18] and [19] the authors apply spectral theory of operators toanalyze BRW on periodic graphs. For this reason they additionally assume the symmetryof the random walk. However, we use other methods based on consideration of our BRW
in the scope of a general BRW with finitely many types of particles. This approach allows
to avoid additional assumption on symmetry of the random walk and permits to consider
the random walk having, for example, a drift. Our proof introduces specified auxiliary
stochastic process and essentially relies on paper [20] devoted to general BRW.
References:
1. Z.Shi. Branching random walks, École d’Été de Probabilités de Saint-Flour XLII – 2012,
Lecture Notes in Mathematics, vol. 2151, 2015.
2. E.Brunet, A.D.Le, A.H.Mueller, S.Munier. How to generate the tip of branching random
walks evolved to large times. Europhysics Letters, Vol. 131, No 4, 2020, 40002.
3. T.Bai, Y.Hu. Capacity of the Range of Branching Random Walks in Low Dimensions.
Proceedings of the Steklov Institute of Mathematics, Vol. 316, 2022, pp. 26–39.
4. E.Chernousova, O.Hryniv, S.Molchanov. Branching random walk in a random timeindependent environment. Mathematical Population Studies, Vol. 30, No 2, 2023, pp. 73–
94.
5. E.Filichkina, E.Yarovaya. Branching Random Walks with One Particle Generation
Center and Possible Absorption at Every Point. Mathematics, Vol. 11, No 7, 2023, pp. 1–
16.
6. V.Vatutin, V.Topchii, E.Yarovaya. Catalytic branching random walk and queueing
systems with random number of independent servers. Theory Probab. Math. Statist, No 69,
2004, pp. 1–15.
7. M.Cranston, L.Koralov, S.Molchanov, B.Vainberg. A continuous model for
homopolymers. Journal of Functional Analysis, Vol. 256, No 8, 2009, pp. 2656–2696.
8. T.Bai, P.Rousselin. Branching random walks conditioned on particle numbers. Journalof Statistical Physics, Vol. 185, No 3, 2021, pp. 1–15.
9. S.S.Bocharov. Fluctuations of the Rightmost Particle in the Catalytic Branching
Brownian Motion. Proceedings of the Steklov Institute of Mathematics, Vol. 316, 2022,
18

pp. 72–96.
10. Rl.Liu. The Spread Speed of Multiple Catalytic Branching Random Walks. Acta
Mathematicae Applicatae Sinica, English Series, Vol. 39, No 2, 2023, pp. 262–292.
11. S.Albeverio, L.Bogachev. Branching random walk in a catalytic medium. I. Basicequations. Positivity, Vol. 4, No 1, 2000, pp. 41–100.
12. L.Doering, M.Roberts. Catalytic branching processes via spine techniques and renewal
theory; C.Donati-Martin, A.Lejay, A.Rouault (Eds.), Séminaire de Probabilités XLV,
Lecture Notes in Mathematics, Vol. 2078, 2013, pp. 305–322.
13. S.Molchanov, E.Yarovaya. Branching processes with lattice spatial dynamics and afinite set of particle generation centers. Doklady Mathematics, Vol. 86, No 2, 2012, pp. 638–
641.
14. Ph.Carmona, Y.Hu. The spread of a catalytic branching random walk. Annales del’Institut de Henri Poincaré, Probabilité et Statistiques, Vol. 50, No 2, 2014, pp. 327–351.
15. E.Vl. Bulinskaya. Spread of a catalytic branching random walk on a multidimensionallattice. Stochastic Processes and their Applications, Vol. 128, No 7, 2018, pp. 2325–2340.
16. E.Vl. Bulinskaya. Maximum of catalytic branching random walk with regularly varying
tails. Journal of Theoretical Probability, Vol. 34, No 1, 2021, pp. 141–161.
17. E.Vl. Bulinskaya. Catalytic branching random walk with semi-exponential increments.
Mathematical Population Studies, Vol. 28, No 3, 2021, pp. 123–153.
18. M.V.Platonova, K.S.Ryadovkin. On the mean number of particles of a branchingrandom walk on Z d with periodic sources of branching. Doklady Mathematics, Vol. 97,
No 2, 2018, pp. 140–143.
19. M.V.Platonova, K.S.Ryadovkin. Branching random walks on Z d with periodic
branching sources. Theory of Probability and its Applications, Vol. 64, No 2, 2019, pp. 229–
248.
20. J.D.Biggins. How fast does a general branching random walk spread; K.B.Athreya,
P.Jagers (Eds.), Classical and Modern Branching Processes, IMA Volumes inMathematics and its Applications, Vol. 84, 1997, pp. 19–40.

Recent results on eigenvalues for branching processes
and related fields
Cloez, Bertrand
UMR MISTEA, INRAE Montpellier
Montpellier cedex 1, France

Law of large number type result for branching processes is related to the long time
behavior of some non-conservative semi-group. We will present some new results forsuch semigroup in the particular case of non-local branching and deterministic dynamics
between branching. These results are based on eigen-problem associated to some integrodifferential operators. The techniques will be related to Doeblin and Lyapunov arguments.
Finally several examples coming from applications in biology will be developed (evolution
models, growth fragmentation...).

19

References:
1. Bansaye, V., Cloez, B., Gabriel, P. Ergodic Behavior of Non-conservative
Semigroups via Generalized Doeblin’s Conditions. Acta Appl Math 166, 29?72 (2020).
https://doi.org/10.1007/s10440-019-00253-5
2. Bansaye, V., Cloez, B., Gabriel, P. and Marguet, A. (2022), A nonconservative Harris ergodic theorem. J. London Math. Soc., 106: 2459-2510.
https://doi.org/10.1112/jlms.12639
3. Cloez, B. (2017). Limit theorems for some branching measure-valued processes.
Advances in Applied Probability, 49(2), 549-580. doi:10.1017/apr.2017.12
4. Cloez, B., Gabriel, P. ErgodicFast, slow convergence, and concentration in the houseof cards replicator-mutator model. To appear in Differential Integral Equations
5. Cloez, B., de Saporta, B., Roget, T. Long-time behavior and Darwinian optimality
for an asymmetric size-structured branching process. J. Math. Biol. 83, 69 (2021).
https://doi.org/10.1007/s00285-021-01695-y.

Lower Large Deviations of Strongly Supercritical
Branching Process in Random Environment with
Geometric Number of Descendants: Local Asymptotics
Denisov Konstantin
Steklov Mathematical Institute of Russian Academy of Sciences
denisovkonstan@yandex.ru

Keywords: branching processes, random environment, random walk, Cramer’s condition,
large deviations, local theorems.
Let η = (η1 , ...) be a sequence of independent identically distributed (i.i.d.) randomvariables (r.v.). Let {φy }y∈R be a family of generating functions. For fixed η we consider
independent random variables Xi,j , j ∈ N, with generating functions φηi , i ∈ N, for eachi ∈ N. We define the branching process (Zn , n ≥ 0) in random environment η (BPRE) by
the following formula:
Z0 = 1,

Zn+1 = Xn+1,1 + · · · + Xn+1,Zn , n ≥ 0.

Let ξi = ln φ0ηi (1) , Eξi = µ. The random walk Sn = ξ1 + · · · + ξn , n ≥ 1, is called theassociated random walk.
We suppose that the step of the associated random walk has positive mean µ andsatisfies left-hand Cramer’s condition E exp(hξi ) < ∞ as h− < h < 0, where h− is someparameter, and that for every i, j r.v. Xi,j have geometric distribution. Also we supposethat ξi is non-lattice r.v. Under these assumptions we consider local probabilities of lower
deviations for BPRE. In other words, we study the local probabilities P (Zn = bexp (θn)c)
as n → ∞ and θ ∈ (max(m− , 0); µ) for some constant m− .
Large deviations of BPRE are well studied. The asymptotic behavior of P(Zn >exp(θn)) as n → ∞, where θ > µ, for BPRE with geometric distribution of the numberof descendants of one particle (BPREG) was studied by Kozlov ([1], [2]). In the generalcase, both the large devation principle ([3]) and the exact asymptotics ([4], [5]]) were20

obtained. For the probabilities of lower deviations P(1 ≤ Zn < exp(θn)), where θ < µ,only logarithmic asymptotic representation was obtained ([6]).
In this report we consider local lower large deviations of BPREG. In other words, westudy P(Zn = k), where k(n) = k ∈ N. We assume that θ(n) = θ := ln k/n belongs
to some segment [θ1 ; θ2 ] ⊂ (max(m− , 0); µ). Under these assumptions we introduce two
deviation zones: the first deviation zone — (max(m(−1), 0); µ), and the second deviation
zone — (max(m− , 0); m(−1)), where m(−1) is some parameter. We show that
1 + o(1) −Λ(θ)n−θn
P (Zn = k) = √eΓ (1 + hθ ) EVe∞hθ −12πnσ (hθ )as n → ∞ uniformly in θ ∈ [θ1 ; θ2 ] where θ1 , θ2 belongs to the first deviation zone, andP (Zn = k) = (1 + o(1)) Rn (−1)EVb∞−2
as n → ∞ uniformly in θ ∈ [θ1 ; θ2 ] where θ1 , θ2 belongs to the second deviation zone ([8]).
Here Ve∞ and Vb∞ are some r.v. and hθ , R(n) and Λ(θ) are some functions. Moreover, we
prove that
P (Zn = k) = (1 + o(1)) ×
 2

√

σ (hθ )n(1 + hθ )2n(θ − m(−1))
hθ −1 −Λ(θ)n−θne×EV∞ eexp
1−Φ2σ(hθ )
as n → ∞ uniformly in θ ∈ [θ1 (n); θ2 (n)] ⊂ (max(m− , 0); µ) such that θ1 (n) → m(−1) andθ2 (n) → m(−1) as n → ∞. Thus, we obtain the asymptotic representation of P (Zn = k)
as n → ∞ that is uniform in θ ∈ [θ1 ; θ2 ] ⊂ (max(m− , 0); µ).
References:
1. Kozlov M.V. On large deviations of branching processes in a random environment:
geometric distribution of descendants.Discrete Mathematics and Applications, Vol. 16,
No 2, 2006, pp. 155-174.
2. Kozlov M.V. On large deviations of strictly subcritical branching processes in arandom environment with geometric distribution of progeny.Theory of Probability and
Its Applications, Vol. 54, No 3, 2009, pp. 424-446.
3. Bansaye V. and Berestycki J. Large deviations for branching processes in random
environment.Markov Process. Related Fields, Vol. 15, No 3, 2009, pp. 493-524.
4. Buraczewski D., Dyszewski P. Precise large deviation estimates for branching process
in random environment.arXiv e-prints, arXiv:1706.03874 , 2017.
5. Shklyaev A. V. Large deviations of branching process in a random environment.
II.Diskretnaya Matematika, Vol. 32, No 1, 2020, pp. 135-156.
6. V. Bansaye, C. Boinghoff. Lower large deviations for supercritical branching processes
in random environment.Proceedings of the Steklov Institute of Mathematics, Vol. 282, No
1, 2013, pp. 15-34.
7. Denisov K. Yu. Asymptotical local probabilities of lower deviations for branchingprocess in random environment with geometric distributions of descendants.Diskretnaya
Matematika, Vol. 32, No 3, 2020, pp. 24-37.
8. Denisov K. Yu. Local lower deviations of strictly supercritical branching process inrandom environment with geometric number of descendants.Diskretnaya Matematika,Vol. 34, No 4, 2022, pp. 14-27.

21

On reduced processes starting from a large number ofparticles
Formanov Shakir, Khusanbaev Yakubdjan
Uzbekistan Academy of Sciences, V.I.Romanovskiy Institute of Mathematics, Tashkent,
100170. Uzbekistan.
yakubjank@mail.ru

Keywords: subcritical and critical branching processes, reduced process, weak convergence.
Let {Zk , k ≥ 0} the Galton-Watson branching process (see e.g., [1]) in which thenumber of direct descendants of one particle have a generating function f (s) , 0 ≤ s ≤ 1.
Denote by Z (m, n) the number of particles in the moment m (m ≤ n) in theprocess {Zk , k ≥ 0}, whose descendants exit at the moment n. The random process{Z (m, n) , 0 ≤ m ≤ n} is called the reduced process generated by the process {Zk , k ≥ 0}.
The reduced process {Z (m, n) , 0 ≤ m ≤ n} is called subcritical, critical and supercriticalif f 0 (1) < 1, f 0 (1) = 1 and f 0 (1) > 1 respectively. Reduced subcritical processes for
Galton-Watson processes were introduced by Fleischmann and Prehn [2] . Fleischmann
and Sigmund-Schultze [3] proved a functional limit theorem (under the assumption
Zn > 0) in which the convergence of reduced critical processes to the Yule process isestablished. Liu and Vatutin [4] proved conditional limit theorems (under the assumption
0 < Z0 ≤ ψ(n) ) for reduced critical processes starting with a single particle and with afinite variance in the number of direct descendants of a single particle.
In this report, we propose limit theorems for subcritical and critical reduced processes{Z (m, n) , 0 ≤ m ≤ n} in the case when Z0 = ϕ (n) with probability 1, where ϕ (n) suchthat ϕ (n) ∼ n or ϕ (n) = o (n) when n → ∞.
We present one of our results.
Theorem. Let for a reduced critical process
0 < f 00 (1) = σ 2 < ∞and with probability 1 Z0 = [xn], where is x > 0 a fixed number, the sign [a] means theinteger part of the number a. Then for any t ∈ [0, 1) the asymptotic relation holds
2x 1−s− ·E sZ([nt],n) / Z0 = [xn] → e σ2 1−st , n → ∞.
(1)
If t = 0, then in the limit (1) a Poisson distribution arises with the parameter σ2x2 . So
the distribution of the number Z0 = [xn] of initial particles that have a descendant in then-th generation at the n → ∞ approaches the Poisson law. This effect is understandable
due to the similarity of the law of development of initial particles.
References:
1. Athreya K., Ney P. Branching Processes. Berlin, Germany: Springer-Verlag. 1972.
2. Fleischmann K., Prehn U. Ein Grenzwertsatz f"ur subkritische Verzwei gungsprozesse
mit eindlich vielen Typen von Teilchen. Math. Nachr., 64 (1974), 357–362.
3. Fleischmann K., Siegmund-Schultze R. The structure of reduced critical Galton–Watsonprocesses. Math. Nachr., 79 (1977), 233–241.
4. Liu M., Vatutin V., Reduced processes for small populations. Theory Probab. Appl.,
63:4 (2018)22

Explosions and dualities in logistic continuous state
branching processes
Foucart, Clément
Université Sorbonne Paris Nord (LAGA) and Ecole Polytechnique (CMAP)
Keywords: branching processes with competition, stochastic duality, explosion, local time
Call Z the process of the size of a random continuous-state population in which classicalreproduction of individuals, governed by a branching mechanism Ψ, (with possibly gigantic
number of offsprings !) is counterbalanced by a quadratic competition drift term: “ −c 2Z dt ”, somehow resulting from duel fights between individuals: two individuals fight2 t
and one kills the other (logistic competition term). These processes were introduced by
A. Lambert in 2005, see [3], who baptized them Logistic CSBPs (LCSBPs).
I will explain in the talk two duality relationships and some of their consequences.
By exploring first a Laplace duality between LCSBPs Z and certain generalized Feller
diffusions U :Z

Laplace dual←→U

we will see that the boundary 0 of the diffusion process U is regular absorbing if and only
if the boundary ∞ of the LCSBP Z is regular reflecting (the process leaves and returns
immediately to ∞ without spending in it a second). Explicit necessary and sufficientconditions on the branching mechanism and the competition term are found for this tohold and the Laplace duality is used to build the extension of the LCSBP, after the first
explosion, when it exists. This part is taken from [1].
In order to study further the process past explosion and for instance to identify its local
time, we are going to introduce another auxiliary process, V , obtained as the Siegmund
dual of the process U :Siegmund dualU←→V
The process V is a certain bi-dual process of Z and many nice connections between Z
and V can be explored, see [2].
References:
1. C. Foucart, Continuous-state branching processes with competition: duality andreflection at infinity, (Electron. J. Probab. 24 (2019), no. 33)
2. C. Foucart, Local explosions and extinction in continuous-state branching processes
with logistic competition, arXiv:2111.06147
3. A. Lambert, The branching process with logistic growth, Ann. Appl. Probab. 15 (2005),
no. 2

23

Large Deviation for Supercritical Controlled Branching
Processes
González, M.1 , Minuesa, C.2 , del Puerto, I.3 , Vidyashnakar, A.N.4
1,2,3

Department of Mathematics. University of Extremadura. Badajoz, Spain.
2
Department of Statistics. George Mason University. Fairfax, USA.
1

mvelasco@unex.es, 2 cminuesaa@unex.es, 3 idelpuerto@unex.es, 4 avidyash@gmu.edu

Keywords: controlled branching processes; supercritical case; large deviations
The topic of large deviation estimates plays an important role for researching manyquestions in statistics. In particular, in the field of the branching processes the study oflarge deviations for standard Bienaymé-Galton-Watson branching processes (BGWPs),{Zn }n≥0 , was initiated in [1] and [2] and investigated in detail in [3] and [4].
Among others results, the large deviation behavior of the statistic Rn = Zn+1 Zn−1 has
been studied. This statistic has been used in the estimation of the amplification rate ina quantitative polymerase chain reaction (PCR) experiment where only Zn and Zn+1 areobserved. For a BGWP, it is known that {Zn+1 Zn−1 }n≥0 converges almost surely to m,
denoting by m the offspring mean, when m > 1 (supercritical case) on the non-extinction
set. In the quoted papers previously, the rate of decay as n → ∞ of P ZZn+1
>

−
m
n
was studied. In particular, the reproductive property of a BGWP was used to link thelarge deviations of Zn−1 Zn+1 with the rates of convergence of the generating function via
the use of the harmonic moments of Zn . The harmonic moments of these processes was
carried out in [5].
A control branching process (CBP) is a generalization of Byenaimé–Galton–Watsonprocesses where at each generation the number of progenitors is randomly chosen througha random control function.
In this talk we present large deviation results for supercritical controlled branchingprocesses under an assumption on the exponential moments or polynomial moments of
the offspring distribution and also based on the asymptotic behaviour of the harmonic
moments of the generation sizes.
Acknowledgements: This research has been supported by grant PID2019-108211GBI00 funded by MCIN/AEI/10.13039/501100011033, by ERDF A way of making Europe.
References:
1. Athreya, K.B., Vidyashankar, A. Large Deviation Results for Branching Processes.
In: Cambanis, S., Ghosh, J.K., Karandikar, R.L., Sen, P.K. (eds) Stochastic Processes.
Springer, New York, NY, 1993.
2. Vidyashankar, A. N. Large deviation results for branching processes in fixed andrandom environments, Thesis (Ph.D.)–Iowa State University, 1994.
3. Athreya, K. B. Large deviation rates for branching processes. I. Single type case,Ann. Appl. Probab. 3, 1994, 779–790.
4. Athreya, K. B. and Vidyashankar, A. N. Large deviation rates for supercritical andcritical branching processes, Classical and modern branching processes, IMA Vol. Math.
Appl. 84, 1-18, 1997.
5. Ney, P. E. and Vidyashankar, A. N. Harmonic moments and large deviation rates
for supercritical branching processes, Ann. Appl. Probab. 13, 2003, 475–489.
24

Conditional central limit theorem for critical andsubcritical branching random walkWenming Hong
Beijing Normal University
wmhong@bnu.edu.cn

Consider a branching random walk on R. Let Zn (A) be the number of the individuals
in the n-th generation located in A ∈ B(R) , and Nn := Zn (R) denote the size of n-th
generation, which is a Galton-Watson process. It is well know that when EN1 = m > 1,
the central limit behavior have been considered by Kaplan and Asmussen (1976) andBiggins (1990). For the critical and subcritical case, we obtained the behavior as follows,(1) when EN1 = m = 1, we prove that, under some conditions, for all x ∈ R, as
n → ∞, (n)√Z (−∞, nx]
LNn > 0 =⇒ L (Y (x)) ,nwhere ⇒ means convergence in law and Y (x) is a random variable whose distribution is
specified by its moments.
(2) when 0 < EN1 = m < 1, we prove that for x ∈ R, under some conditions, as
n → ∞,
√L Zn ((−∞, nx]) | Nn > 0 =⇒ L(ξ1{N ≤x} ),where ⇒ means convergence in law, ξ is the Yaglom limit of the subcritical GaltonWatson process {Nn ; n ≥ 0} conditioned on non-extinction, N is a standard normal
random variable and independent of ξ .
This is a joint work with Shengli Liang and Dan Yao.
Keywords: branching random walk, subcritical Galton-Watson process, critical
Galton–Watson process, reduced process, Yaglom’s limit, conditional central limit theorem.
References
1. Biggins, J. D. (1990). The central limit theorem for the supercritical branchingrandom walk, and related results. Stochastic Process. Appl. 34, 255-274.
2. Fleischmann, K. and Siegmund-Schultze, R. (1977). The structure of reduced critical
Galton-Watson processes. Math. Nachr. 79, 233-241.
3.Fleischmann, K., Vatutin, V.(1999). Reduced subcritical Galton-Watson processes ina random environment. Adv. in Appl. Probab. 31, 88-111.
4.Harris, S. C. and Roberts, M. I. (2017). The many-to-few lemma and multiple spines.
Ann. Inst. Henri PoincarГ c Probab. Stat. 53, 226-242.
5.Hong, W.M., Liang, S.L.(2023+). Conditional central limit theorem for criticalbranching random walk. Priprint.
6.Hong, W.M.,Yao, D.(2023+). Conditional central limit theorem for subcriticalbranching random walk. Priprint.
7.Kaplan, N. and Asmussen, S. (1976). Branching random walks. II. Stochastic
Process. Appl. 4, 15-31.

25

On explicit expression of the Generating Function ofInvariant Measures of Critical Galton-Watson
Branching Systems
Azam A. Imomov
Karshi State University, Karshi, Uzbekistan
imomov_ azam@mail.ru

Keywords: Galton-Watson Branching System, Generating functions, Slow variation,
Basic Lemma, Transition probabilities, Invariant measures, Limit theorems, Convergencerate.
Let N be the set of natural numbers and N0 = {0} ∪ N. Consider an ordinary GaltonWatson Branching (GWB) system with a state space S0 ⊂ N0 and an offspring law{pj , j ∈ S0 }. Let Z(n) be the population size at the moment n ∈ N0 . The stochastic
system {Z(n)} forms a reducible, homogeneous and discrete-time Markov chain whosestate space consists two classes: S0 = {0} ∪ S, where S ⊂ N, therein {0} is an absorbing
state, and S is the class of possible essential communicating states. The offspring law{pk , k ∈ S} fully defines a structure of the GWB system. In fact, we observe that ani
appropriate probability generating function (GF) E sZ(n) Z(0) = i = fn (s) for alls ∈ [0, 1), where the GF fn (s) = E1 sZ(n) is n-fold functional iteration of GF
X
f (s) :=
pk sk .
k∈S0

Denoting q be the smallest root of the fixed-point equation f (s) = s for s ∈ [0, 1], werecall that fn (s) → q as n → ∞ uniformly in s ∈ [0, r] for any fixed r < 1. So, the GWB
system is a discrete dynamic system generated by the GF f (s) and with the fixed point
q, which is an extinction probability of a trajectory of the system initiated by a single
founder; see [1, Ch. I].
We consider a case when the offspring GF f (s) for s ∈ [0, 1) admits the following form:


1
1+ν
f (s) = s + (1 − s) L
,
[fν ]
1−s
where 0 < ν < 1 and L(∗) slowlyPvaries [2] at infinity. Assumption [fν ] implies that the
per-capita offspring mean m := j∈S jpj = f 0 (1−) = 1 and f 00 (1−) = ∞, so that oursystem is critical type with infinite variance. In the case when the slowly varying at zerofunction L(∗) replaces L(∗) in [fν ], Slack [4] has shown that there exists an invariant
measure whose GF U (s) has the following local expression:
U (s) ∼1ν(1 − s)ν L(1 − s)as s ↑ 1.

In this report we provide an alternative argument against Slack’s one and we obtain
the global expression for all s ∈ [0, 1) of the function U (s) and its derivative. Let
V(s) :=1νΛ(1 − s)and26J(s) :=

1 − f 0 (s)− 1,Λ(1 − s)where Λ(y) := y ν L (1/y).
Theorem 1. If p0 > 0 and the condition [fν ] is satisfied, then(i) the GF U (s) has the following form:
U (s) = V(s) − V(0);(ii) the derivative U 0 (s) has the following expression:
U 0 (s) = J(s)

V(s)
.
1−s

P
Remark. The function U (s) admits the power series expansion U (s) = j∈S uj sj ,PPwhere uj = k∈S uk Pkj (1) and k∈S uk pk0 = 1; see [4, Lemma 4]. Then it follows thatu1 = U 0 (0) =J(0)1 − p0 − p 1
.
=
νp0
νp20

The assertions of the Theorem improve the corresponding results from [3].
References:
1. Athreya, K. B. and Ney, P. E. Branching processes. Springer, New York, 1972.
2. Bingham, N. H., Goldie, C. M. and Teugels, J. L. Regular variation, Cambridge, 1987.
3. Imomov, A. A. and Tukhtaev, E. E. On asymptotic structure of critical Galton-Watson
branching processes allowing immigration with infinite variance. Stochastic Models,39(1), 118–140, DOI: 10.1080/15326349.2022.2033628. 2023.
4. Slack, R. S. A branching process with mean one and possible infinite variance.
Wahrscheinlichkeitstheor. und Verv. Geb. 9, 139–145, 1968.

On refinement of some limit theorems for thenoncritical Galton-Watson Branching Systems
Azam Imomov, Misliddin Murtazaev
Karshi State University, Karshi, Uzbekistan
Romanovskiy Institute of Mathematics, Tashkent, Uzbekistan
imomov_ azam@mail.ru, misliddin1991@mail.ru

Keywords: Galton-Watson Branching System, Markov chain, Extinction time,
Kolmogorov constant, Basic Lemma, Limit Theorems, Invariant Distribution,
Convergence rate.
Let Z(n) be a population size at the moment n ∈ N0 in the Galton-Watson branching(GWB) system with branching rates pk , k ∈ N0 , where N0 = {0}∪N and N = {1, 2, . . .}.
This is a reducible, homogeneous-discrete-time Markov chain with a state space consistingof two classes: S0 = {0} ∪ S, where {0} is absorbing state, and S ⊂ N is the classof possible essential communicatingstates. We assume throughout this paper that 0 <P
p0 + p1 < 1 and m := k∈S kpk < ∞. The parameter m is the average number of direct
descendants of one individual in one-step generation. We are interested in the subcritical27

and supercritical cases, which are assigned m < 1 and m > 1 respectively. Denoting q
be an extinction probability of the system initiated by a single founder, we recall that
it is smallest nonnegative root of the fixed-point equation f (s) = s on the domain of{s : s ∈ [0, 1]}, whereXf (s) =
pk sk
k∈S0

is the offspring generating function (GF). The extinction probability q = 1 in subcriticalcase, and q < 1 when the system is supercritical. So, the supercritical system survives
with positive probability; see [1].
Put into consideration n-step transition probabilities
Pij (n) := P Z(n + k) = j Z(k) = i
for any k ∈ N0 .
Pi
A corresponding probability GF k∈S0 Pij (n)sk = fn (s) , whereX
fn (s) :=pk (n)sk ,k∈S0

therein pk (n) := P1k (n). Then fn (0) = p0 (n) is a vanishing probability of the system
initiated by a single founder. This probability tends monotonously to q as n → ∞, i.e.
limn→∞ p0 (n) = q. Furthermore fn (s) → q as n → ∞ uniformly in s ∈ [0, 1); see [1, Ch.I.].
Let Rn (s) := q −fn (s). Denoting H := min {n ∈ N : Z(n) = 0} be an extinction
time of
the system initiated by a single founder, we note that Q(n) := Rn (0) = P n < H < ∞is the probability of that the system survives at the time n but degenerates eventually.
For the subcritical case P {H < ∞} = 1 and, hence Q(n) = P {Z(n) > 0}. In this case,
Kolmogorov [2] proved that if
f 00 (1−) < ∞ for m < 1[K]then Q(n) admits the following asymptotic representation:Q(n) = Kmn 1 + o(1)as n → ∞,(1)

where K is the well-known Kolmogorov constant, but it does not have an explicit form
here.
This report aims to improve and generalize the asymptotic formula (1) to thenoncritical case. Denote
f 00 (q).
β := f 0 (q)
and
γq :=
2β (1 − β)Theorem 1. Let m 6= 1. If Kolmogorov condition [K] is satisfied,i) then
P n<H<∞
n
=
1
−
B
K
β
1
+
o(1)
as n → ∞,qq
Kq β n(2)ii) andq 
E Z(n) n < H < ∞ =1 + Bq Kq β n 1 + o(1)
Kq
where
Kq =

q
1 + qγq

and
28

Bq =

f 00 (q).
2β

as n → ∞,(3)

Remark. The principal novelties of Theorem 1 are as follows. First, it generalizes andasymptotically refine Kolmogorov’s result (1) and the analogous theorem, established by
Sevastyanov [3] only for the subcritical case. Secondly, the main terms on the right-handside of both statements (2) and (3) involve an explicit form of the constant Kq . Finally,
the decrease rates decay of the second term in these asymptotic expansions are found.
Theorem 2. Let m 6= 1 and the condition [K] is satisfied. Then the following
asymptotic relation holds:
p1 (n)
1 2 
n
1
+
o(1)
as n → ∞,
=
K
·
1
+
2γ
K
β
q q
βn
q2 q
where Kq is the Kolmogorov constant appearing in Theorem 1.
References:
1. Athreya, K. B. and Ney, P. E. Branching processes. Springer, New York, 1972.
2. Kolmogorov A. N. K resheniyu odnoy biologicheskoy zadachi. Reports of SRI Math.
and Mech. at Tomsk Univ. 2, pp. 7–12, 1938. (Russian)
3. Sevastyanov B. A. Branching processes. Nauka, Moscow, 1971. (Russian)

Periodic branching processes with immigration andtheir implicit multi-type representation
Ispány Márton; Bondon Pascal; Reisen V.A.
University of Debrecen, Faculty of Informatics, Department of Information Technology,
Hungary
Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes,
FranceUniversidade Federal de Minas Gerais, Department of Statistics, Brazil
ispany.marton@inf.unideb.hu, pascal.bondon@centralesupelec.fr, valderioanselmoreisen@gmail.com

Keywords: branching process with immigration, periodicity, implicit representation, YuleWalker method.
Nowadays, there is a growing interest in non-Gaussian time series, particularly inseries comprised of non-negative integers or counts, see the latest survey [1]. Count seriesarise in fields, such as agriculture, economics, epidemiology, finance, geology, meteorology,
and sports. Many natural or human phenomena exhibit periodic behavior. A potentiallypromising model to describe periodic count time series is the periodic non-negative integervalued autoregressive and moving average (PINARMA) model based on the binomial
thinning operation, see [2]. A particular periodic and seasonal example of the PINARMA
model is considered in [3].
In this talk, the generalized PINARMA model with period S ∈ N, autoregressive orders(ps ) ∈ NS0 and moving average orders (qs ) ∈ NS0 is considered which is defined as
YkS+s =psX

αk,s,i ◦ YkS+s−i +

i=1

qs
X
j=1

29

βk,s,j ◦ εkS+s−j + εkS+s ,k ∈ Z, s = 1, . . . , S, where {αk,s,i ◦ | k ∈ Z} and {βk,s,j ◦ | k ∈ Z} are sequencesof identically distributed generalized thinning operators with non-negative means αs,i ,i = 1, . . . , ps and βs,j , j = 1, . . . , qs , s = 1, . . . , S, respectively, and the immigration
sequence {εkS+s | k ∈ Z, s = 1, . . . , S} consists of N0 -valued random variables.
The parameters αs,i ’s and βs,j ’s are called periodic autoregressive and moving averagecoefficients. We denote by YkS+s the series during the sth season of period k. We
P recallthat the generalized thinning operator α◦ is defined as the random sum α ◦ Y := Yj=1 αj ,where {αj | j ∈ N} is a sequence of independent identically distributed N0 -valued
random variables with non-negative finite mean α, and Y is a N0 -valued random variablewhich is mutually independent of {αj }. All thinning operators αk,s,i ◦’s and βk,s,j ◦’s aresupposed to be mutually independent and independent of the immigration process {εt }.
The generalized PINARMA model can be interpreted as a higher-order branching processwith immigration in varying environment.
We show that the generalized PINARMA model can be written, with the help of thegeneralized matricial thinning operator, in a S-dimensional stationary vector model form,
which is called the VINARMA model, as
Yk =

p
X

Ak,i ◦ Y k−i +

i=0

q
X

Bk,j ◦ εk−j ,j=0where Y k := (YkS+S , YkS+S−1 , . . . , YkS+1 )> and εk := (εkS+S , εkS+S−1 , . . . , εkS+1 )> , k ∈ Z,(> denotes transpose). In this equation, the entries of the matricial thinning operators
Ak,i ◦’s and Bk,j ◦’s are defined with the help of the thinning operators αk,s,i ◦’s and βk,s,j ◦’s,and the autoregressive and moving average orders are p := [max1≤s≤S (ps − s)/S] + 1 andq := [max1≤s≤S (qs − s)/S] + 1, respectively, where [x] denotes the greatest integer lessthan or equal to a real x. The sequences {Ak,i ◦ | k ∈ Z} and {Bk,j ◦ | k ∈ Z} consistof identically distributed generalized matricial thinning operators with mean matrices Ai
and Bj , respectively, for all i = 0, 1, . . . , p, j = 0, 1, . . . , q. Ai ’s and Bj ’s are non-negativesquare matrices of dimension S. In particular, A0 is strictly upper triangular, and B0
is upper triangular with unit diagonal. If A0 6= 0 then the VINARMA model is called
proper implicit since Y k appears on both sides of the equation. We suppose that {εk } isa sequence of independent identically distributed random vectors and P(ε = 0) < 1.
An essentially optimal and simple spectral criterion based on the model parameters isgiven for the existence and uniqueness of a solution to the above stochastic models. Let
ρ(M ) denote the spectral radius of a matrix M . Let Gk denote the σ-algebra generated
by the matricial thinning operators Al,i ◦, i = 0, 1, . . . , p, Bl,j ◦, j = 0, 1, . . . , q, and r.v.’s
εl , l ≤ k, for all k ∈ Z. {Y k } is called a non-anticipative solution to the VINARMA
model if the set of random variables {Y j | j ≤ k} are mutually independent of the setof matricial thinning operators {Al,i ◦, Bl,j ◦ | l > k, i = 0, 1, . . . , p, j = 0, 1, . . . , q} and theimmigration vectors {εj | j > k} for all k ∈ Z.
Theorem 1. Suppose that Ekε0 k < ∞ and the matrices {A0 , A1 , . . . , Ap } satisfy
ρ(A0 + A1 + . . . + Ap ) < 1. Then the VINARMA model has a unique non-anticipative
solution {Y k } which can be expressed as the almost sure convergent infinite series
Yk =

∞
X

(j)

Zk ,

j=1
(n)

k ∈ Z, where Z k denotes the number of nth generation offspring of immigrants at time30

k. The NS0 -valued stochastic process {Y k } is {Gk }-adapted, ergodic and strictly stationary
pth order homogeneous Markov chain with finite mean.
We provide a complete description of the probabilistic structure, among others, themean and the covariance function of unique solutions to the PINARMA and VINARMA
models. Two infinite series representations, moving average and immigrant generation,
are also derived. A successive approximation procedure based on immigrant generation
representation is proposed for constructing the unique solution to the models, which canbe used to simulate the process efficiently, see [5]. The Yule-Walker method is applied
to estimate the parameters of the PINARMA and VINARMA models. To investigate
the asymptotic behavior of Yule-Walker estimators, we prove limit theorems extending
some known results of periodic multi-type Galton-Watson branching process without
immigration, see [4].
Acknowledgments
This research was supported by the Project no. TKP2020-NKA-04 which has been
implemented with the support provided from the National Research, Development andInnovation Fund of Hungary, financed under the 2020-4.1.1-TKP2020 funding scheme.
References:
1. R.A. Davis, K. Fokianos, S.H. Holan, H. Joe, J. Livsey, R. Lund, V. Pipiras, N.
Ravishanker. Count time series: A methodological review. Journal of the American
Statistical Association, Vol. 116, 2021, pp. 1533-1547.
2. M. Bentarzi, N. Aries. QMLE of periodic integer-valued time series models.
Communications in Statistics - Simulation and Computation, Vol. 51, No 9, 2022, pp.
4973-4999.
3. P.R.P. Filho, V.A. Reisen, P. Bondon, M. Ispany, M.M. Melo, F.S. Serpa. A periodic andseasonal statistical model for non-negative integer-valued time series with an application
to dispensed medications in respiratory diseases. Applied Mathematical Modelling, Vol.
96, 2021, pp. 545-558.
4. M. Gonzalez, R. Martinez, M. Mota. A note on the asymptotic behaviour of a periodic
multitype Galton-Watson branching process. Serdica Mathematical Journal, Vol. 30, No
4, 2004, pp. 483-494.
5. M. Ispany, P. Bondon, P.R.P. Filho, V.A. Reisen. Existence of a periodic and seasonal
non-negative integer-valued autoregressive process. Submitted to Journal of Time Series
Analysis, 2023.

On refinements of the asymptotic expansion of thecontinuation of critical branching processes
Juraev Shukur
Kokand State Pedagogical Institutejurayev4756@gmail.com

Keywords: Critical branching processes, probability of the continuation.
For the probability of the continuation of critical branching processes, an asymptotic
expansion is obtained under the assumption of the existence of factorial moments αk for
k = 4, 5, ..., m, m < ∞.
31

Let Zn , n = 0, 1, 2, ..., be a branching process with discrete time and one type ofparticles and Qn = 1 − P0 (n) be the probability of the continuation of the process.
The asymptotic behavior of probability Qn for discrete time was studied by A. N.
Kolmogorov [1]. The results of A. N. Kolmogorov for processes with continuous time wereobtained by B. A. Sevastyanov [2].
A literature review on the issues of limit theorems and local limit theorems, and inparticular, refinement of the asymptotic expansion for probability Qn , is briefly presented
in the publication by S. V. Nagaev and R. Mukhamedkhanova [3].
In this abstract, we consider some refinements of the theorems proven in [3] on theasymptotic expansion for probability Qn .
The following theorems were proven in [3] for critical branching processes (A = 1), (see[3], pp. 96-97):
Theorem 1. If A = 1, B > 0, C < ∞, then as n → ∞:




4C
2
2 ln n
ln n
+
(1)
Qn =
−
+o
Bn
3B 3 B n2n2
Theorem 2. If A = 1, B > 0, D < ∞, then as n → ∞:




2
4C
2 ln n
4K
ln n
Qn =
+
−
+ 2 2 +O,
Bn
3B 3 B n2
B n
n3

(2)

where K is some constant dependent on the form of F (x).
In [3], the authors reported that their methods for proving relations (1)–(2) are suitable
for the case when factorial moments of a higher order exist.
Although not significant, The authors also considered the case when there are factorial
moments αk = F (k) (1) < ∞ for k ≥ 4.
Now we proceed to the consideration of the case when there are factorial moments
αk < ∞, where k = 4, 5, ..., m, m < ∞.
Theorem 3. If A = 1, B > 0, αk < ∞, k ≥ 4, then as n → ∞ for Qn the following
asymptotic formula holds :( l 2l+1 l+1 l+1 2i+1 i iX
2
2
L1 ln nL1 ln n
i
Qn =
(−1)
+
+
i+1
l+2
B
n
B
n
i=0
)
" l

 2(i+1) i i #

X
ln n
2
L1 ln ni+1+(−1)
(Km + Mnm )
1+o
(3)
B
ni+2
n
i=0
where m ≥ 4,
Km = 1 + L1

2c1
1+
+O
B

  m−2
m−2
X
X
1
+
Lj (1 + Ij ) −Lj ζ (j) ,
n
j=2
j=2
∞

m−2
X

B2 C
2 X1
Mnm =
Lj Rnj , Rnj =
(1
+
o
(1))
,
L
=
−
1
B t=n tj
4
6
j=2
Ij =

∞
X
t=1

Qjt ,

ζ (j) =

∞
X
1

tj
t=1
32

− Eyler zeta f unction,and coefficients Li , i = 1, 2, ..., m − 1, depend only on factorial moments αk < ∞, k =2, 3, ..., m, c1 = 0, 577216...− is the Eyler’s constant, and parameter l = 1, 2, ..., is defined
below.
Comment
Parameter l in formula (3) determines the number of steps in the process of division
with a remainder. From a practical point of view, it is more advantageous to assume
that l = 1 or 2, or 3, etc. Obviously, parameter l allows us to determine the number ofasymptotic terms in the asymptotic expansion for Qn the probability of continuation ofcritical branching processes with discrete time.
It is easy to see that the expansion (3) contains (l + 1) (m − 1) + 1, (m ≥ 4) of
asymptotic terms, each of which has an explicitly defined constant coefficient, depending
only on factorial moments αk < ∞, and the independent parameters l and m take thevalues of l = 1, 2, ..., m = 4, 5, ....
References:
1. Kolmogorov A.N. To the solution of one biological problem. "News of Research Instituteof Mathematics and Mechanics, Tomsk University 1938, V.2, issue 1.
2. Sevastyanov B.A. Branching processes M.: Nauka, 1971, 603 p.
3. Nagaev A.V., Mukhamedkhanova R., Some limit theorems from the theory of branchingprocesses. Limit theorems and statistical inferences., Tashkent: Fan, 1966, pp. 90-112.
4. Vatutin V. A., Branching processes and their applications. V. A. Steklov Mathematical
Institute, RAS. Lecture courses REC. Issue 8. Moscow 2008. 107 p.

Explosion phenomena for Fleming-Viot-type processes
Martin Kolb
We will consider branching processes of Fleming-Viot-type which are driven by Bessel
type diffusions on the non-negative reals with drift pointing towards the origin. This
process has been previously considered by Burdzy et al., where the authors have beenable to fully characterize, when the two particle system is well-defined. In the case of morethan two particles a sufficient condition for the process to be well-defined was derived. In
the talk we recall these results and present a new abstract critierium, which in the caseof three particles sharpens the previous results significantly.

33

A limit theorem for the critical Galton-Watson
branching processes
Kudratov Khamza, Khusanbaev Yakubdjan
Samarkand State University named after Sharof Rashidov, 15, University street, 100174,
Samarkand, Uzbekistan,e-mail:qudratovh− 83@mail.ru

Institut of Mathematics, Tashkent, Uzbekistan,e-mail:yakubjank@mail.ru

Keywords: Critical Galton-Watson process, generating function, slowly varying function.
Suppose that {ξ(k, j) , k, j ∈ N} be a sequence of independent identically distributed
random variables taking non-negative integer values. Let the random variable ξ(1, 1) havethe distribution
pk = P (ξ(1, 1) = k) , k = 0, 1, ...,
with the generating functionξ(1,1)

F (s) := Es=∞X

pk sk , 0 ≤ s ≤ 1,

k=0

and p0 + p1 6= 1. Consider the process W (k), k ≥ 0 defined by the following recurrent
relation:
W (n−1)X
W (0) = η , W (n) =ξ(n, j) , n ∈ N,(1)
j=1

here η is a random variable that takes positive integer values and independent on thesequence of random variables{ξ(k, j) , k, j ∈ N} .
We call the process {W (k) , k ≥ 0} the Galton-Watson process starting with arandom number of particles η. It is well known [1], the asymptotic state of the process{W (k) , k ≥ 0} depends on the mean value of the random variable ξ(1, 1), and it isdivided into the classes as follows. It is clear that F 0 (1) = Eξ(1, 1). The process (1)is called subcritical, critical and supercritical if F 0 (1) < 1, F 0 (1) = 1 and F 0 (1) > 1,
respectively.
In this work, we consider only critical processes.
We denote the Galton-Watson process generated by the i-th particle in the initialstate by Wi (n), n = 0, 1, .... Obviously, Wi (n), n = 0, 1, ... , i ≥ 1 form independent andidentically distributed Galton-Watson branching processes. It is known [1] that W (n) canbe represented asηX
W (n) =Wi (n) , n ∈ N.
(2)
i=1

34

Independence of random variables η and ξ(i, j), i ≥ 1, j ≥ 1 implies independence ofWi (n) and the random variable η. Denote by P (n) the probability of degeneration of
the process {W (k) , k ≥ 0} at the n-th step, i.e. P (n) = P (W (n) = 0). We denote byR(n) the probability of continuation of the process W1 (n) at the n-th step, i.e. R(n) =P (W1 (n) > 0). In what follows, we need the following designations:
Q(n) = 1 − P (n) , h(s) := Esη , Hn (s) := EsW (n) , A = h0 (1) , σ 2 = F 00 (1) ,F0 (s) = s, F1 (s) = F (s), Fn (s) = F (Fn−1 (s)) is the n-th iteration of F (s).
Further the sign an ∼ bn indicates that lim abnn = 1.
n→∞

In 1968, Slack [3] considered the case of
F (s) = s + (1 − s)1+α L (1 − s) , α ∈ (0, 1] ,(S)

here L(x) is a slowly varying function on a neighborhood of zero, and obtained thefollowing:1(1 − Fn (0))α L(1 − Fn (0)) ∼,(3)αnlim E (exp {−λ(1 − Fn (0))W (n)} /W (n) > 0) = 1 − λ (1 + λα )−1/α , λ > 0.

n→∞

(4)

This result implies the result by Yaglom [2] if α ≡ 1 and F 00 (1) < ∞. It should be notedthat in the case considered by Slack, the equality F 00 (1) = ∞ can be satisfied.
In [4], K.V. Mitov, G.K. Mitov, N.M. Yanev considered the critical case (F 0 (1) = 1)
when the second factorial moment was finite: F 00 (1) = σ 2 < ∞, and the generating
function of the number of particles in the initial state was satisfied the condition1θ
h(s) = 1 − (1 − s) L0, θ ∈ (0, 1),(M )1−shere L0 (x) is a slowly varying function at infinity, and obtained the following results:
P (W (n) > 0) = 1 − h(Fn (0)) ∼ (σ 2 n)−θ L0 (n),(5)

lim E (exp {−λ(1 − Fn (0))W (n)} /W (n) > 0) = 1 − λθ (1 + λ)−θ , λ > 0.

(6)

n→∞

With the help of Tauber’s theorems, it is not difficult to see that condition (M ) implies
that the average number of particles in the initial state is infinitely. But it follows from (7)that in this case, too, the critical Galton-Watson process will degenerate with probability
1.
We got the following result.
Theorem. If the conditions (M) and (S) are satisfied, thenlim E (exp {−λ(1 − Fn (0))W (n)} /W (n) > 0) = 1 − λθ (1 + λα )−θ/α , λ > 0.

n→∞

In the case of F 00 (1) < ∞, Theorem implies the result by Mitov, Mitov, and Yanev. If
we set formal θ = 1, α = 1 in the last Laplace substitution, we get the Laplace substitution(1 + λ)−1 of the exponential distribution.
References35

1. Harris T.E. The theory of branching processes. Springer - Verlag, Berlin-GöttingenHeidelberg, 1963. 355p.
2.Yaglom A. M: Nekotorie predelnie teoremы teorii vetvyashchixsya sluchaynix
protsessov. DAN SSSR, 1947, 56, 8, 795-798.(in Russian)
3. Slack R.S A Branching Process with Mean One and Possibly Infinite Variance. Z.
Wahrsch. Verw. Gebiete. 9, 139-145 (1968).
4. Mitov K.V., Mitov G.K., Yanev N.M. Limit theorems for critical randomly indexedbranching processes. Workshop on Branching Processes and Their Applications. Springer,2010.
5. Kudratov Kh.E., Khusanbaev Ya.M. Some limit theorems for the critical GaltonWatson branching processes. Ukrainian mathematical journal. Vol 75 No 4 (2023).

A periodic branching random walk with immigration
on Zd
Lukashova Irina
St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of
Sciences
ilukashova072@gmail.com

Keywords: Branching random walks, immigration, periodic potential, direct integral
decomposion, asymptotic behavior.
We consider a continuous-time branching random walk with immigration on Zd with
branching sources located periodically. We assume that the processes of branching andimmigration are independent and the intensity matrices of random walk of immigration
centers and particles are denoted by {ai (v, u)}v,u∈Zd , i = 0, 1 respectively. The intensityof branching and immigration at a point v are denoted by β(v) and α(v) respectively.
Let Ai and Q be the operators l2 (Zd ) → l2 (Zd ) such thatXai (v, u)f (u), i = 0, 1,(Ai f (·))(v) =u∈Zd

Qf (v) = β(v)f (v).
During the talk, the equation for the mean number of particles at the moment t at
the point u with starting point v will be presented and the following theorem will be
formulated:
Theorem 1. Let λ1 (0) be the upper bound of the spectrum of the operator A1 + Q.
1. If λ1 (0) > 0, then for every u, v ∈ Zd there exists C = C(u, v, d) such that 
eλ1 (0)t1
M (v, u, t) = C(v, u, d) · d/2
1+O, t → ∞.
tt
2. If λ1 (0) < 0, then for every u, v ∈ Zd there exists C̃ = C̃(u, v, d) such that

 
1
1
m0 (v, u, t) = C̃(v, u, d) · d/2 1 + O, t → ∞.
tt

36

References:
1. Reed M., Simon B. Methods of modern mathematical physics, v. 4., Elsevier, 1978.
2. Sevastyanov B.A. Limit theorems for branching stochastic processes of special form.
Theory Probab Appl., v.2(3), 1957, 339 - 348.
3. Platonova M.V., Ryadovkin K.S. A branching random walk on Zd with branchingsources located periodically. Theory Probab. Appl., v.64:2, 2019, 229 - 248.

Asymptotic fluctuations of supercritical generalbranching processes
Matthias Meiners

The Crump-Mode-Jagers (CMJ) process is a fairly general branching process thatunifies and extends earlier models of individual-based branching processes. Nerman’scelebrated law of large numbers (1981) states that, for a supercritical CMJ process (Zt )t≥0 ,under some mild assumptions, e−αtZt converges almost surely as t → ∞ to aW. Here,α > 0 is the Malthusian parameter, a is a constant and W is the limit of Nerman’s
martingale, which is positive on the event that the population survives.

Time reversal of spinal processes for linear andnon-linear branching processes near stationarity
Méléard Sylvie
Ecole Polytechnique et Institu,France
We consider a stochastic individual-based population model with competition, trait
structure affecting reproduction and survival, and changing environment. The changes oftraits are described by Gaussian or jump processes, and the dynamics can be approximatedin large population by a non-linear PDE with a non-local mutation operator. Using thefact that this PDE admits a non-trivial stationary solution, we can approximate the nonlinear stochastic population process by a linear birth-death process where the interactions
are frozen, as long as the population remains close to this equilibrium. This allows us toderive, when the population is large, the equation satisfied by the ancestral lineage of anindividual uniformly sampled at a fixed time T, which is the path constituted of the traitsof the ancestors of this individual in past times before T.
This process is a time inhomogeneous Markov process, but we show that the time
reversal of this process possesses a very simple structure (e.g. time-homogeneous andindependent of T).
37

Bayesian inference in controlled branching processes
via ABC methodology
Minuesa Carmen, González Miguel and del Puerto Inés
Department of Mathematics, University of Extremadura, Badajoz, Spain cminuesaa@unex.es,
mvelasco@unex.es, idelpuerto@unex.es,

Keywords: approximate Bayesian computation, Bayesian inference, controlled branchingprocesses, logistic growth population model.
In a Bayesian setting, we aim at estimating the posterior distribution of the parametersof interest in controlled branching processes without determining the exact likelihood
function or prior knowledge of the maximum number of offspring that an individual cangive birth.
To that end, we have developed approximate Bayesian computation (ABC) algorithms
for branching processes. More precisely, to estimate the maximum number of children per
individual we present an ABC rejection method for model choice based on the comparison
with the observed raw data. In the next stage, using an appropriate summary statistic weapproximate the posterior distributions of the target parameters by employing a tolerancerejection method along with a post-sampling correction.
We illustrate the methodology by means of some examples developed with thestatistical software R.
References:
1. González, M., Minuesa, C., del Puerto, I. Approximate Bayesian computation
approach on the maximal offspring and parameters in controlled branching processes.
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A.
Matemáticas, 116, 4, 2022, 147. doi:10.1007/s13398-022-01290-w

The total progeny in the positive recurrent Q-processes
Zuhriddin A. Nazarov
V.I.Romanovskiy Institute of Mathematics, Tashkent 100174, Uzbekistan
zuhrov13@gmail.com

Keywords: Branching system, Q-process, Markov chain, extinction time, total progeny,
positive recurrent.
Let {Z(n), n ∈ N0 } GWB system with branching rates {pk , k ∈ N0 }, where N ={1, 2, . . .}, N0 = {0}∪N and the variable Z(n) denote the population size at the moment n
in the system. The evolution of the system occurs according to the following mechanism.
Each individual lives a unit length life time and then gives k ∈ N0 descendants withprobability pk . This process is a reducible, homogeneous-discrete-time Markov chain witha state space consisting of two classes: S0 = {0} ∪ S, where {0} is absorbing state, andS ⊂ N is the class of possible essential communicating states. Throughout the paper
38

assume that p0 > 0 andP p0 + p1 > 0 which called the Schröder case. We suppose thatp0 + p1 < 1 and m := k∈S kpk < ∞.
Considering transition probabilities
Pij (n) := P Z(n + k) = j Z(k) = i
for any k ∈ N0
we observe that the corresponding probability generating function (GF)X
i
Pij (n)sk = fn (s) ,

(1)

k∈S0

P
where fn (s) := k∈S0 pk (n)sk , therein pk (n) := P1k (n) and fn (s) is n-fold iteration ofP
the offspring GF f (s) := k∈S0 pk sk . Needless to say that fn (0) = p0 (n) is a vanishingprobability of the system initiated by one individual. Note that this probability tends
as n → ∞ monotonously to q, which called an extinction probability of the system, i.e.
limn→∞ p0 (n) = q. The extinction probability q = 1 if m ≤ 1 and q < 1 if m > 1. Basedon this, according to the values of the parameter m, the system is called sub-critical if
m < 1, critical if m = 1 and super-critical if m > 1.
Further we are dealing with the GWB system conditioned on the event {n < H < ∞},where
H := min {n ∈ N : Z(n) = 0}is the extinction time. Let Pi ∗ := P ∗ Z(0) = i and define conditioned probability
measureH(n+k){∗} := Pi ∗ n + k < H < ∞Pi
for any k ∈ N.
In [2] proved, that
H(n+k) 

Qij (n) := lim Pik→∞Z(n) = j =jq j−iPij (n),
iβ n

(2)

P
where β := f 0 (q). Observe that j∈N Qij (n) = 1 for each i ∈ N. Thus, the probability
measure Qij (n) can determine a new population growth system with the state space E ⊂N which we denote by {W (n), n ∈ N0 }. This is a discrete-homogeneous-time irreducibledMarkov chain defined in [2] and called the Q-process. Undoubtedly W (0) = Z(0) andtransition probabilities
Qij (n) := P W (n) = j W (0) = i = Pi Z(n) = j H = ∞ ,so that the Q-process can be interpreted as a “long-living” GWB system. Put intoP(i)consideration a GF wn (s) := j∈E Qij (n)sj . Then from (1) and (2) we obtainwn(i) (s) =

fn (qs)q

i−1· wn (s),

(3)




(1)
where the GF wn (s) := wn (s) = E sW (n) W (0) = 1 has a form of wn (s) = sfn0 (qs) β n
for all n ∈ N. Using iterations for f (s) in (3) leads to the following functional equation:(i)wn+1 (s) =w(s) (i)wn fq (s) ,fq (s)
39(4)


where w(s) := w1 (s) and fq (s) = f (qs) q. Thus, Q-process is completely defined by
setting the GF
f 0 (qs).
w(s) = s(5)βAn evolution of the Q-process is in essentially regulated by the structural parameterβ > 0. In fact, as it has been shown in [2], that E is positive recurrent if β < 1 and E istransient if β = 1. On the other hand, it is easy to be convinced that positive recurrent
case β < 1 of Q-process corresponds to the non-critical case m 6= 1 of the initial GWB
system. Note that β ≤ 1 and nothing but.
In this paper, we deal with the positive recurrent case assuming that first moment α :=0w (1−) be finite. Then differentiating (5) on the point s = 1 we obtainh α = 1 + γq (1 −iβ),00where γq := qf (q) β (1 − β). It follows from (3) that Ei W (n) := E W (n) W (0) = i =
(i − 1) β n + EW (n), where EW (n) = 1 + γq 1 − β n .
It is obvious, that when initial GWB system is sub-critical, then the condition α < ∞is this is equivalent to that f 00 (1−) < ∞. Further we everywhere will be accompanied bythis condition by default. Our purpose is to investigate asymptotic properties of a randomvariable
Sn = W (0) + W (1) + . . . + W (n − 1),denoting the total number of individuals that have existed up to the n-th generationin Q-process. By analogy with branching systems, this variable is of great interest in
studying the deep properties of the Q-process. Our main results are analogues of central
limit theorem and law of large numbers for Sn .
Theorem 1. Let β < 1 and α < ∞. Then
Sn − ESn→ N0,σ2Kn

as n → ∞,where N0,σ2 – a√normal distribution function with zero mean and finite variance of σ 2 > 0
and Kn := O∗ ( n).
Theorem 2. Let β < 1 and α < ∞. Then there exists slowly varying function at
infinity L(∗) such thatSn − ESnL(n)P< x − N0,σ2 ≤ 1/4
Knn
uniformly in x.

Theorem 3. Let β < 1 and α < ∞. Then the distribution of Sn n converges weakly
to the degenerate distribution concentrated at the point 1 + γq , i.e.

if x ≤ 1 + γq ,
 0,
Sn
P
< x ⇒ I1+γq (x)andI1+γq (x) = 1,nif x > 1 + γ .
qMoreover there exists slowly varying function at infinity Lγ (∗) such that
Lγ (n)
SnP< x − I1+γq (x) ≤ √nnuniformly in x.
40

References:
1. Asmussen S. and Hering H. Branching processes. Birkhäuser, Boston, 1983.
2. Athreya K. B. and Ney P. E. Branching processes. Springer, New York, 1972.
3. Harris T. E. The theory of branching processes. Springer-Verlag, Berlin, 1963.
4. Sevastyanov B. A. Branching processes. Nauka, Moscow, 1971. (Russian)Gaussian waves in BBM with mean-dependent
branching
Sarah Penington
University of BathWe consider a continuous-space analogue of a population model introduced by Yu,
Etheridge and Cuthbertson. We prove a hydrodynamic limit result that allows us toshow that for a large total population size, at large times the empirical distribution of
the particle positions evolves approximately according to an accelerating Gaussian wave.
Based on joint work with Erin Beckman.

Functional limit theorems and the asymptotic
normality of estimators based on partial observations
Rahimov Ibrahim, Sharipov Sadillo.
V.I.Romanovskiy Institute of Mathematics, Tashkent, Uzbekistan
ibrakhimjonrakhimov@gmail.com, sadi.sharipov@yahoo.com

Keywords: Branching process; immigration; restricted observation; offspring mean.
The asymptotic normality of estimators of the offspring mean is important for
applications. In the case when the population sizes are partially observed the limit
theorems for the process are not sufficient to obtain asymptotic distributions for
known estimators. As a result the asymptotic normality of the estimators based onpartial observations has not been obtained in its standard form and leads to consider
various modifications of the estimators. In the talk, we demonstrate that the functional
limit theorems for the critical partially observed process with generation-dependentimmigration allow to show that the original (non-modified) estimators are asymptoticallynormal. For this we first extend known functional limit theorems for fully observed
processes obtained by Wei and Winnicki [1] and Rahimov [2], respectively, to the caseof partial observations which is of independent interest as well. Then, we use the newtheorems to obtain desired asymptotic normality.
References:
1. Wei, C. Z., Winnicki, J. Some asymptotic results for branching processes withimmigration.Stochastic Processes and their Applications, Vol. 31, No 2, 1989, pp. 261282.
2. Rahimov I. Functional limit theorems for critical processes with immigration. Adv.
Appl. Probab., Vol. 39, No 4, 2007, pp. 1054-1069.
41

Branching process for the solution of semi-linear
Helmholtz boundary value problem
Rasulov Abdujabbor, Raimova Gulnora
University of World Economy and Diplomacy, Tashkent
asrasulov@gmail.com, raimova27@gmail.com

Keywords: Helmholtz equation, Dirichlet problem, Monte Carlo method, branching random
process, walk on spheres, martingale, unbiased estimator.
As is known, there are many applied problems, the solution of which is associated withboundary value problems for nonlinear elliptic equations containing hyperbolic functionsof unknown functions. For example, the equation ∆u(x) = k 2 · ch(u) arises in problemsof constructing surfaces of constant mean curvature in the hyperbolic space H 3 [1]. The
equations ∆u(x) = k 2 · sh(u) arise when solving problems of bio-molecular electrostatictheory [2]. This article considers a probabilistic approach to solving the first boundary
value problem for the following two below equations. Let D be a bounded domain in R3
with regular boundary Γ, ϕ(x) ∈ C(Γ̄), ψ(x) ∈ C(Γ̄). Consider the following Dirichletproblems:−∆u(x) + c · u(x) = g · sh(u), x ∈ D, u|Γ = ϕ;−∆u(x) + c · u(x) = g · ch(u),x ∈ D,

u|Γ = ψ.

It is assumed that the functions ϕ(x), ψ(x) and the coefficients c, g are such that there
exists a unique continuous solution to these semilinear problems ([3], [4]). Assuming theexistence of a solution for the problems, an unbiased estimator is constructed on the
trajectories of the branching process walk on spheres. Unbiased and biased estimators forthe solution of boundary value problems for the linear Helmholtz equation ∆u − cu =−g(x) were considered for c(x) = const by G.A. Mikhailov and B.S. Elepov in [5], [6],for the variable case c(x) in [7], [8] for the Dirichlet problem, N.A. Simonov in [9] for themixed problem and the problem Neumann, A.S. Sipin in [10] for the Dirichlet problem
for the equation
n
X
∂u
∆u +
ai
+ au = −g.
∂xii=1In the papers [11], [12] by G.A. Mikhailov and R.N. Makarov, estimates for the solutionof boundary value problems for the linear Helmholtz equation are built on the basis of a
special integral-difference equation using the process of walking on spheres with reflection
from the border. In the works [13], [14] the Monte Carlo solution of one applied problem of
biomolecular electrostatic theory for the linearized equation ∆u(x) = k 2 ·u was considered.
The Monte Carlo solution of the Dirichlet problem for a nonlinear equation of the form∆u(x) =nXai (x)u2i (x) + a0 (x)i=1

was proposed by A.S.Rasulov in his work [15], [16]. G.A. Mikhailov in [7] studied special
case the equation ∆u + un = 0 and in [17], [18] G.M.Raimova applied for the equation∆u(x) + c u(x) =

∞
X
i=0

42

ai (x)ui (x).

In these work we will study a probabilistic representation of the solution of theHelmholtz boundary problem for the non-linear problem−∆u(x) + cu(x) = g · f (u),x ∈ D,

u|Γ = ψwhere, f (u) in our case could be hyperbolic functions sh(u) or ch(u). Under theassumption of the existence of a solution, an unbiased estimator is constructed on the
trajectories of the proposed branching process "walk on spheres". To do this, using Green’s
formula, a special integral equation is written that connects the value of the function with
its integrals over a ball and a sphere of maximum radius centered at a point and entirelycontained in the region under consideration. It is proved that under certain conditions
there exists a fixed point for the nonlinear integral operator corresponding to the integral
equation. In this case, the iteration process method converges and classical Monte Carlo
methods could be used. A probabilistic representation of the solution of the problem
in the form of the mathematical expectation of some random variable is obtained. In
accordance with the probabilistic representation, a branching process of walk on spheresis constructed and an unbiased estimator of the solution of the problem with finite varianceis constructed on its trajectories.
References:
1. A.I.Bobenko. All constant mean curvature tori in R3 , S 3 , H 3 in terms of theta-functions.
Mathematische Annalen, 290.2 (1991): 209-246., http://eudml.org/doc/164816
2. M.Davis, A.McCammon. Electrostatics in-molecular bio-structure and dynamics.
Chemical Reviews, 90 (1990), pp. 509-521.
3. R.Courant, D.Hilbert. Methods of Mathematical Physics: Partial Differential Equations,Wiley-VCH, Volume 2, 1991, p.852.
4. Yu.A.Dubinsky. Quasilinear elliptic and parabolic equations of any order. RussianMathematical Surveys. - Moscow, 1968. - N.23, issue. 1(139). pp.45-90.
5. B.S.Elepov, G.A.Mikhailov. Algorithm of "wandering over spheres"for the equation.
Doklady mathematics, Moscow, 1973. - T.212, No. 1. pp.15-18.
6. B.S.Elepov, G.A.Mikhailov. Using fundamental solutions of elliptic equations toconstruct algorithms for the Monte Carlo method. Computational Mathematics andMathematical Physics. Moscow, 1974. V.14, No.3. pp.728-736.
7. G.A.Mikhailov. Solving the Dirichlet problem for nonlinear elliptic equations by theMonte Carlo method. Siberian Mathematical Journal. Novosibirsk, 1994. - T.35, No. 5.
pp.1085-1093.
8. A.S.Sipin. Solution of the Dirichlet Problem for the Equation −∆u(x) + a(x)u = f (x)by Monte Carlo Methods. Vestnik of Leningrad State University, Ser. Math.,Mech., Astr.
v.1,(1976), 60-63 (in Russian)
9. N.A.Simonov. Algorithms for random walk over spheres for solving a mixed boundary
value problem and the Neumann problem. Numerical Analysis and Applications,
Novosibirsk, 2007. - T. 10, No.2, pp. 209-220.
10. A.S.Sipin. Solution of two Dirichlet boundary value problems by the Monte Carlo
method. Computational Mathematics and Mathematical Physics, Moscow, 1979, V.19,
No.2. pp.388-401.
11. G.A.Mikhailov, R.N.Makarov. Solving boundary value problems of the second andthird kind by Monte Carlo methods. Siberian Mathematical Journal, Novosibirsk, 1997,
V.38, No.3. pp.603-614
12. G.A.Mikhailov, R.N.Makarov. Solution of boundary value problems by the method of43

"walking on spheres"with reflection from the boundary. Doklady mathematics, Moscow,
1997, T.353, N6, pp.720-722.
13. M.Fenley, M.Mascagni, J.McClain, A.Silalahi and N.Simonov. Using correlated Monte
Carlo sampling for efficiently solving the linearized Poisson-Boltzmann equation over abroad range of salt concentration. J. Chem. Theory Comput. 6 (2009), 300-314.
14. C.Fleming, M.Mascagni and N.Simonov. An efficient Monte Carlo approach forsolving linear problems in bio-molecular electrostatics. Computational Science-ICCS,
2005, Springer, 2005, pp. 760-765.
15. A.S.Rasulov. Monte-Carlo Method for Solving Nonlinear Problems. Monograph,
Tashkent, Fan, 1992, p.105 (in Russian).
16. A.S.Rasulov. Monte Carlo Algorithms for the Solution Quasi-Linear Dirichlet
Boundary Value Problems of Elliptical Type, Mathematics and Statistics, Horizon
Research Publishing corporation, 2023, vol.11, No.2, p.200-205
17. G.M.Raimova. Probabilistic models for solving the Dirichlet problem for nonlinear
elliptic equations. Uzbek mathematical journal, Tashkent, 2017. - No. 1, pp. 114-123.
18. G.M.Raimova. Probabilistic Approach to Solution of the Neumann Problem for Some
Nonlinear Equation. Communications in Statistics - Simulation and Computation,2016,
Volume 45, Issue 8, pp.2981-2990.

On random walks in random environment
with random local constraints
Sakhanenko Alexander
Sobolev Institute of Mathematics, Novosibirsk, Russia
aisakh@mail.ru

Keywords: conditioned random walk, bounded local times, regenerative sequence, potential
regeneration, separating levels, skip-free distributions, accompanying process.
Consider a d-dimensional random walk
St = (St [1], . . . , St [d]) = S0 +

t
X

ξj ,

t = 0, 1, 2, . . . ,j=1

on the integer lattice Zd , where ξj = (ξj [1], . . . , ξj [d]) ∈ Zd , j = 1, 2, . . . , are i.i.d. random
vectors that do not depend on the initial value S0 ∈ Zd . We assume that at any time
t = 0, 1, 2, . . . the number of possible/allowed visits to each state x ∈ Zd is limited above
by a counting number Ht (x) ≥ 0. Let
T∗ = inf{t ≥ 0 : Ht (St ) = 0} ≤ ∞be the first time when the walk visits a state with zero number of possible/allowed visits
to it. If T∗ is finite, we assume that the random walk “freezes” at the time instant T∗ (orit “dies”, or “is killed” at time T∗ ).
We assume also that, at any time t < T∗ , the random walk jumps from St−1 to St andchanges the environment at point St−1 by decreasing the number of remaining allowed
44

visits by 1, so thatHt (x) := Ht−1 (x) − 1{St−1 = x} for each x ∈ Zd

and all 0 ≤ t ≤ T∗ .

Thus, we consider a multidimensional integer-valued random walk in a changing random
environment.
As a natural example, consider a model of a random walk on atoms of a “harmonic
crystal”. An electron jumps from one atom to another, taking from a visited atom for the
next jump a fixed unit of energy, that cannot be recovered. Thus, if St is a position of
the electron at time t, then it takes a unit of energy to make the next jump to position
St+1 = St + ξt+1 , which may be in any direction from St since the ξ’s are signed randomvariables. When the electron arrives at an atom with insufficient energy level, it “freezes”
there.
We interpret the first coordinate St [1] of St as its height and assume further that theheight cannot increase by more than one unit:ξt [1] ≤ 1 a.s.,t = 1, 2, . . . .

(1)

Under the skip-free property (1) we may define the hitting time α(n) of the level n:α(n) := inf{t ≥ 0 : St [1] ≥ n} = inf{t ≥ 0 : St [1] = n}.
Our simplest result is that, under natural technical assumptions (see detailes in [1] or [2]),there exist positive constants 0 < q∞ ≤ 1 and 0 < c0 < ∞ such that we have the following
relation:nP(Bn ) ∼ c0 q∞
as n → ∞,where Bn := {α(n) < T∗ }.
(2)
Thus in (2) we have found an exact asymptotic for the probability of the event Bn that
our random walk reaches the level n before it “freezes”.
Secondly, we prove convergence of the conditional distributions:
P((S0 , . . . , SK ) ∈ A | Bn ) → P((S 0 , . . . , S K ) ∈ A),

as n → ∞,(3)

for any k = 0, 1, 2, . . . and all A ⊂ Z(K+1)×d , where Z(K+1)×d denotes the space of
vectors ~x = (x0 , x1 , . . . , xK ) having d-dimensional vectors as their components. Further,
we find that the limiting sequence {S k } in (3) has a regenerative structure with an infinite
sequence of random regenerative levels {ν i } and increases to infinity with a linear speed,i.e.
S n [1]/n → a1 ∈ [0, 1] a.s.
as n → ∞.
(4)
Our proofs of results (2) – (4) in [1] and [2] are based on establishing a number ofrepresentations for the distribution of the random walk {St } in random environment{Ht (x)}, that is linked to the distribution of the limiting sequence {S t }. For example, it
is shown in [2] that, under some technical assumptions,nP(Bn ) = ψ0 q∞
P(B n ),

where B n := ∪nm=0 {ν m = n},(5)

for a well-defined positive constant ψ0 and for q∞ as in (2); and that
P((S0 , . . . , SK ) ∈ A | Bn ) = P((S 0 , . . . , S K ) ∈ A | B n ),(6)

for any n ≥ K = 0, 1, 2, . . . and all A ∈ Z(K+1)×d . Here event B n occurs if (and only if)
the given number n is one of the regenerative levels of the limiting random walk.
45

In paper [3], we have found that the limiting process is not the only one for whichsuch representations do exist. We showed that there exist random sequences {S t }, thatare called “accompanying” sequences and that depend on the used below numbers n andq, and are such that
P(Bn ) = ψn (q)q n P(B n )for all q ≥ qn > 0,(7)where 0 < qn ≤ 1 and 0 < ψn (q) < ∞ are well-defined constants. We have to underlinethat representation (7) may take place also in cases then formulas (2) – (6) do not holdbecause the number q∞ does not exist.
Several generalizations of results from [2] and [3] will be presented in the talk.
Earliar, in [4], convergences (2) – (4) were proved in a particular case whenP(ξ1 = 1) = 1/2 = P(ξ1 = −1),

S0 = 0 and H0 (x) = L0 = const ≥ 2

for all x ∈ Z. The latter means that initially each atom has a fixed (the same for all)amount of energy L0 . Paper [4] has motivated us to introduce and study the generalized
model.
References:
1. A. Sakhanenko and S. Foss. On the structure of a conditioned random walk on the
integers with bounded local times. Siberian Electronic Mathematical Reports, Vol. 14,
2017, pp. 1265-1278.
2. S. Foss and A. Sakhanenko. Structural Properties of Conditioned Random Walks on
Integer Lattices with Random Local Constraints. Progress in Probability, Springer, Vol.
77, 2021, pp. 407–438.
3. A. Sakhanenko and S. Foss. On Representations and Simulation of Conditioned Random
Walks on Integer Lattices. Siberian Electronic Mathematical Reports, Vol. 18, No 2, 2021,
pp. 1556-1571.
4. I. Benjamini and N. Berestycki. Random paths with bounded local time. Journal of theEuropean Mathematical Society, Vol. 12, No 4, 2010, pp. 819–854.

Pushed and pulled waves in population genetics
Emmanuel Schertzer
This talk is motivated by the stochastic F-KPP equation with Allee Effectr11u(1 − u)η∂t u = ∂xx u + u(1 − u)(1 + Bu) +2
N
where η is a space-time white noise. Numerical results and heuristics by Physicists suggest
the existence of an interesting phase transition between a pulled, a semi pushed and a fully
pushed regime. First, I will start with a brief explanation of the three regimes and their
biological implications. I will then introduce a toy model which mimics the qualitative
behavior of the aforementioned model. This is a class of critical branching Brownian
motions with inhomogeneous branching rates which can be treated analytically using
recent methods which are interesting on their own (moments of random trees, k-spines).
This is joint work with J. Tourniaire (University of Vienna/ISTA) and Felix Foutel–Rodier(Oxford).
46

Asymptotics for the site frequency spectrum associatedwith the genealogy of a birth and death process
Schweinsberg Jason, Shuai Yubo
University of California at San Diego
Keywords: Birth and death process, Coalescent point process, Site frequency spectrum.
Consider a birth and death process started from one individual in which each individual
gives birth at rate λ and dies at rate µ, so that the population size grows at rate r = λ−µ.
Lambert [1] and Harris, Johnston, and Roberts [2] came up with methods for constructing
the exact genealogy of a sample of size n taken from this population at time T . We use
the construction of Lambert, which is based on the coalescent point process, to obtain
asymptotic results for the site frequency spectrum associated with this sample. In the
supercritical case r > 0, our results extend results of Durrett [3] for exponentially growing
populations. In the critical case r = 0, our results parallel those that Dahmer and Kersting[4] obtained for Kingman’s coalescent.
References:
1. Lambert, A. The contour of splitting trees is a Levy process. Ann. Probab., 38(1), 2010,
pp. 348–395.
2. Harris, S. C., Johnston, S. G., Roberts, M. I. The coalescent structure of continuoustime Galton-Watson trees. Ann. Appl. Probab., 30(3), 2020, pp. 1368–1414.
3. Durrett, R. Population genetics of neutral mutations in exponentially growing cancer
cell populations. Ann. Appl. Probab., 23, 2013, pp. 230-250.
4. Dahmer, I., Kersting, G. The internal branch lengths of the Kingman coalescent. Ann.
Appl. Probab., 25(3), 2015, pp. 1325–1348.

Large Deviations of Bisexual Brancing Processes in
Random Environment
Shklyaev Alexander
Steklov Mathematical Institute of Russian Academy of Sciences
ashklyaev@gmail.com

Keywords: Bisexual Branching Process in Random Environment, Processes with
Immigration, Large Deviations, Cramer Condition
Bisexual branching processes (BBP) were introduced by D. Daley in [1]. He consideredS
i.i.d. random vectors (Ui,j , Vi,j ), i, j ∈ N, with N0 × N0 values, were N0 = N {0} anddefined BBP Nn by the equation
!
Nn
Nn
X
X
Nn+1 = min
Un,i ,
Vn,i , n ≥ 0, N0 = 1.
i=1

i=1

47

It corresponds to the simple probabilistic model – we have particles of two sexes, they
form pairs (’mating units’), every pair produce a random vector (X, Y ) of descendants.
This model is called BBP with completely promiscuous mating. In general situation
!
Nn
Nn
X
X
Nn+1 = L
Un,i ,
Vn,i , n ≥ 0,
i=1

i=1

were L : N0 × N0 → N0 is some function. The function L is called the mating function of
the process.
This model was studied by D. Daley, D. Hull, F. Bruss, J. Bagley, M. Gonzalez, M.
Molina. A review of some important results can be found in [2].
We consider bisexual branching processes in random environment (BBPRE) introducedby Ma in [3]. We suppose that there exists a infinite sequence η of i.i.d. r.v. ηi (random
environment), consider mating function L : N0 × N0 × R → N0 and a family of twodimensional N0 × N0 distributions {Lz , z ∈ R}. We assume that for given η• the mating function in the i-th generation is L(·, ·, ηi );• the numbers of descendants (Ui,j , Vi,j ), j ∈ N of the mating units of i-th generation
are i.i.d. random vectors with the distribution Lηi .
For this model in [3] the extinction was studied. We study large deviation probabilities
in this model.
We introduce the condition|L(x, y, z) − g(x, y, z)| ≤ c1 (|x| + |y|)1−δ
for some Lipschitz (with respect to x, y) function g : R+ × R+ × R, satisfying g(cx, cy, z) =cg(x, y, z) for all x, y, c ∈ R+ , z ∈ R. All popular mating functions L satisfy this condition.
Under this condition we introduce
χi = ln Eηi Ui,1 ,ζi = ln Eηi Vi,1 ,ξi = g(χi , ζi , ηi ).

We call the random walk Sn = ξ1 + · · · + ξn the associated random walk for BBPRE Nn .
Under Cramer conditions on ξi and some moment conditions on Ui , Vi we prove that
P(ln Nn ∈ [x, x + ∆)) ∼ I(x/n)P(Sn ∈ [x, x + ∆)),
were I is some function, ∆ > 0 is some constant, x/n ∈ [θ1 , θ2 ] ⊆ (m∗ , m+ ), where m∗ ,m+ are some constants.
We also consider BBPRE with immigration and obtain similar results in this model.
References:
1. Daley, D. Extinction conditions for certain bisexual Galton-Watson branching processes.
Zeitschrift fГjr Wahrscheinlichkeitstheorie und verwandte Gebiete, 9.4, 1968, pp 315-322.
2. Molina, M. Two-sex branching process literature. Workshop on branching processes andtheir applications. Springer Berlin Heidelberg, 2010, pp. 279-293.
3. Ma, Shi-xia. Bisexual Galton-Watson branching processes in random environments.
Acta Mathematicae Applicatae Sinica, 22, 2006, pp. 419-428.

48

Coalescence in Bisexual Branching Processes
Yadav, Sumit Kumar
Indian Institute of Technology Roorkee, Roorkee, India
sumitky@ms.iitr.ac.in

Keywords: Coalescence, Branching Process, Bisexual, Poisson, Mating function.
The study of branching processes has long been recognized as a powerful tool inunderstanding the dynamics of populations and the evolution of species. Bisexual
branching processes provide a valuable framework for investigating scenarios where
reproduction is possible between individuals of different genders. One fundamental aspectof such processes is coalescence, which refers to the merging of ancestral lineages overtime. This study delves into the mathematical modeling of these processes, exploring
the dynamics of lineages across generations and the probability of coalescence events.
Coalescence times, which signify the duration it takes for a pair of lineages to merge,
have implications for understanding the genetic diversity and effective population sizeof species. Moreover, the concept of coalescence provides essential tools for studying
population genetics and phylogenetics. In the literature, a lot of studies have focused on
bisexual branching process. Also, a lot of studies have focused on coalescence problem
in several variants of discrete time Galton Watson Branching Process. However, very
few studies have explored the coalescence problem in Bisexual Branching Process. We
consider a discrete time bisexual branching process and consider a few special cases ofsuperadditive mating functions to obtain interesting theoretical results. Further, using
extensive simulation, we also observe the phenomenon of coalescence by generalizing theprocess to multitype bisexual branching process, where there can be multiple types ofindividuals. Some interesting insights have been obtained using simulation.
References:
1. Hull, David M., A Necessary Condition for Extinction in Those Bisexual GaltonWatson Branching Processes Governed by Superadditive Mating Functions.Journal ofApplied Probability, Vol. 19, No 4, 1982, pp. - 847-850.
2. Casimiro Corbacho, Manuel Molina, Manuel Mota, A mathematical model to describe
the demographic dynamics of long-lived raptor species.Biosystems, Vol. 180, 2019, pp.
54-62 .
3. Chang, Joseph T. , Recent Common Ancestors of All Present-Day Individuals..Advancesin Applied Probability, Vol. 31, No 4, 1999, pp. 1002-1026 .
4. M. GonzГЎlez, C. GutiГ c rrez and R. MartГnez ,Bayesian Inference in Y-Linked
Two-Sex Branching Processes with Mutations: ABC Approach..IEEE/ACM Transactions
on Computational Biology and Bioinformatics, Vol. 18, No 2, 2021, pp. 525-538 .

49

On the rate of convergence in limit theorems forfluctuation critical branching processes withimmigration
Toshkulov Khamza1 , Khusanbaev Yakubdjan2
1

2

Samarkand State University named after Sharof Rashidov, 15, University street,
100174, Samarkand, Uzbekistan,

V.I.Romanovsky Institute of Mathematics of the Academy of Sciences of the Republic of
Uzbekistan.
kh.toshkulov@mathinst.uz, yakubjank@mail.ru

Keywords: Branching process; generating function; weak convergence.
Let {ξk,j , k, j ≥ 1} and {εk , k ≥ 1} be two independent collections of independent,
random variables taking non-negative integer values such that {ξk,j , k, j ≥ 1} identical
distributed. Let the sequence of random variables Xk , k ≥ 0 be defined by the following
recursive relations:
Xk−1
X
X0 = 0, Xk =
ξk,j + εk , k = 1, 2, ...
(1)
j=1

Stochastic processes defined in this way often appear in population theory (see, forexample, [1]) and are called branching processes with immigration.
Let m = Eξ1,1 < ∞. The branching process (1) is called subcritical, critical andsupercritical if m < 1, m = 1 and m > 1 respectively. We obtain estimates a for the rateof convergence of the distribution of fluctuations of a branching process with immigration
to the normal law and the obtained estimates are applied to the case when the immigration
flow {εk , k ≥ 1} is inhomogeneous and the mean value and variance of εn are regular andincreasing.
Let us introduce the following notation: m = Eξ1,1 , σ 2 = Dξ1,1 , γ = E |ξ1,1 − 1|3 < ∞,nnPP
λk = Eεk , b2k = varεk , θk = E |εk − λk |3 , Γn =(εk − λk ), T2n = DΓn , H2n = σ 2
Xk−1 .
An =

n
P

k=1

k=1

λk , Bn2 − some sequence of positive numbers, Φσ (x) −normal distribution

k=1

with mean zero and variance σ 2 , Φ (x) −standard normal distribution,Xn − An∆n = sup P< x − Φ (x) .
Bn−∞<x<∞Let us agree to denote by C, C1 , C2 , ...− positive absolute constants.
Theorem 1. Let m = 1, γ < ∞, b2k < ∞, k ∈ N . Then the following inequality is true

1/4
n
P
3/2
EXk−1 h γ i1/4  k=1∆n ≤ C1 3
3/2  +
n
σ
 P

Ak−1
k=1

50

"
+C2

H2
E n2 − 1Dn#
3/2 1/4


+3

T2n
2πBn2

1/3
+√

Bn
1
−1 .
2πe DnTheorem 2. Let m = 1, 0 < σ 2 < ∞, random variables εk , k ≥ 1 are independentand θk < ∞, k ≥ 1. Then the following inequality is true
 P
1/3
n
2
 2

σ
A
n
 k=1 k−1 
1 X
B
1
n

∆n ≤ C T3
θk + 3  2πB 2  + √2πe T2 − 1 .
n
n
n
k=1

References
1. Haccou P., Jagers P., Vatutin V.A., Branching processes. Variation, growth andextinction of population – Cambridge Univer., Press. 2005. – 317 p.

Random walks conditioned to stay nonnegative andbranching processes in nonfavorable random
environment
Vatutin Vladimir, Dyakonova Elena, Dong Congzao
Steklov Mathematical Institute, Moscow
Steklov Mathematical Institute, Moscow
Xidian University, Xi’an, China
vatutin@mi-ras.ru, elena@mi-ras.ru, czdong@xidian.edu.cn

Keywords: Random walk, branching processes, conditional limit theorems.
Let {Sn , n ≥ 0} be a random walk whose increments belong without centering to thedomain of attraction of an α-stable law {Yt , t ≥ 0}, i.e. Snt /an ⇒ Yt , t ≥ 0, for somescaling constants an . Assuming that S0 = o(an ) and Sn ≤ ϕ(n) = o(an ) → ∞, weprove several conditional limit theorems for the distribution of Sn−m given m = o(n) andmin0≤k≤n Sk ≥ 0. These theorems complement the statements established by F. Caravenna
and L. Chaumont [1].
Let, further, {Z(k), k ≥ 0} be a critical branching process evolving in random
environment and {Sk , k ≥ 0} be its associated random walk. Using the results obtained for
random walks, we continue the study of processes evolving in unfavorable environment
initiated at [2] and [3] and investigate the distribution of the properly scaled process{log Zk , k ≥ 0} for the moments k = n − m given that m = o(n), Zn > 0 andSn ≤ ϕ(n) = o(an ) → ∞.
Acknowledgment. The work of E.E. Dyakonova and V.A. Vatutin was performed at
the Steklov International Mathematical Center and supported by the Ministry of Scienceand Higher Education of the Russian Federation (agreement no. 075-15-2022-265). The
research of C.Dong and V.A. Vatutin was also supported by the Ministry of Science andTechnology of PRC, project G2022174007L.
51

References:
1. Caravenna F. and Chaumont L. An invariance principle for random walk bridges
conditioned to stay positive.Electron. J. Probab., Vol. 18, No 60 , 2013, pp. 1–32.
2. V. Vatutin, E. Dyakonova. Critical branching processes evolving in an unfavorable
random environment. Diskretnaya matematika, Vol.34 , No 3, 2022, pp. 20–33. (InRussian), arXiv:2209.13611
3. V. Vatutin, E. Dyakonova. Population size of a critical branching process evolving in
unfavorable environment. Teoriya Veroyatn Priment., Vol. 68, No 3, 2013, (in Russian, in
print).

Times of a branching process with immigration invarying environment attaining a fixed level
Wang, Hua-Ming
Anhui Normal University, Wuhu, P.R. China
hmking@ahnu.edu.cn

Keywords: Branching processes, varying environments, immigration, regeneration
Consider a branching process {Zn }n≥0 with immigration in varying environment. Fora ∈ {0, 1, 2, ...}, let C = {n ≥ 0 : Zn = a} be the collection of times at which the
population size of the process attains level a. We give a criterion to determine whether
the set C is finite or not. Especially, if a = 0, C is just the set of regeneration times. For
critical Galton-Watson process, we show that |C ∩ [0, n]|/ log n → S in distribution, where
S is an exponentially distributed random variable with P (S > t) = e−t , t > 0.
References
1. Athreya, K.B. and Ney, P.E. Branching processes. Berlin/Heidelberg/New York,
Springer-Verlag, 1972, 287 pages.
2. Csáki, E., Földes, A. and Révész, P. On the number of cutpoints of the transient nearest
neighbor random walk on the line. J. Theor. Probab., Vol. 23, No. 2, 2010, pp. 624-638.
3. Kersting, G. and Vatutin, V. Discrete time branching processes in random environment.
London, ISTE Ltd and Hoboken, John Wiley & Sons, Inc, 2017, 286 pages.
4. Petrov, V.V. A generalization of the Borel-Cantelli lemma. Statist. Probab. Lett., Vol.
67, No. 3, 2004, pp. 233-239.
5. Shiryaev, A.N. Probability, 2nd ed. Berlin/Heidelberg/New York, Springer-Verlag, 1996,
621 pages.

52

A Countable-Type Branching Process Model for the
Tug-of-War Cancer Cell Dynamics
Ren-Yi Wang, Marek Kimmel
Department of Statistics, Rice University, Houston, TX, 77005, USA
rw47@rice.edu, kimmel@rice.edu

Keywords: Multitype branching process, cancer dynamics, negative selection, deleterious
passenger mutations, Tug-of-War.
We consider a time-continuous Markov branching process of proliferating cells witha countable collection of types. Among-type transitions are inspired by the Tug-of-War
process introduced by [1] as a mathematical model for competition of advantageous driver
mutations and deleterious passenger mutations in cancer cells. We introduce a version of
the model in which a driver mutation pushes the type of the cell L-units up, while apassenger mutation pulls it 1-unit down. The distribution of time to divisions depends on
the type (fitness) of cell, which is an integer. The extinction probability given any initialcell type is strictly less than 1, which allows us to investigate the transition between types(type transition) in an infinitely long cell lineage of cells. The analysis leads to the resultthat under driver dominance, the type transition process escapes to infinity, while underpassenger dominance, it leads to a limit distribution. Implications in cancer cell dynamics
and population genetics are discussed.
References:
1. McFarland, Christopher D., Leonid A. Mirny, and Kirill S. Korolev. "Tugof-war between driver and passenger mutations in cancer and other adaptive
processes."Proceedings of the National Academy of Sciences 111.42 (2014): 15138-15143.

Near-critical branching processes considered as Markov
chains with small drift
Wachtel Vitali
Bielefeld University, Bielefeld, Germany
wachtel@math.uni-bielefeld.de

Let Zn be a state-dependent branching process with migration. In the talk I shalldiscuss√an approach to this rather wide class of processes, which is based on transformation
Xn = Zn and which allows one to obtain the results for branching processes without
using generating functions.

53

Growth-fragmentation and quasi-stationary methods
Alex Watson
Abstract: A growth-fragmentation is a stochastic process representing cells with
continuously growing mass and sudden fragmentation. Growth-fragmentations are used
to model cell division and protein polymerisation in biophysics. A topic of wide interest
is whether or not these models settle into an equilibrium, in which the number of cells is
growing exponentially and the distribution of cell sizes approaches some fixed asymptotic
profile. In this work, we present a new spine-based approach to this question, in which acell lineage is singled out according to a suitable selection of offspring at each generation,with death of the spine occurring at size-dependent rate. The quasi-stationary behaviourof this spine process translates to the equilibrium behaviour, on average, of the growthfragmentation. We present some Lyapunov-type conditions for this to hold. This is joint
work with Denis Villemonais (Ecole des Mines de Nancy/Universite de Lorraine).

Spectral methods and their applications in the theoryof branching random walks
Yarovaya Elena
Lomonosov Moscow State University
yarovaya@mech.math.msu.su

Keywords: branching random walks, multidimensional lattices, Green’s functions,martingales, limit theorems.
In the modern theory of stochastic processes and their applications, the martingale
methods are commonly understood as a wide range of probabilistic and analyticaltechniques based on the concept of a martingale (i.e. a process whose prediction in the“future”, which is based on the “past”, depends only on the “current state”). In the talkwe examine an application of the theory of martingale and spectral analytical methods toa number of problems in such an actively developing field of stochastic processes as thetheory of branching random walks (BRWs). Using of BRWs makes it possible to study
the evolution of particle systems that can not only give offspring or die, but also walkon multidimensional lattices under various assumptions on environments according torules that take into account “randomness” of a process see, e.g., [1]–[5]. We propose new
methods for the study of BRWs based on a combination of martingale technique (see,Smorodina and Yarovaya, 2022) and the spectral theory which allowed, firstly, to expand
the class of studied BRWs [6], secondly, to prove new limit theorems on the convergencein the mean square of some functionals, which defined on the trajectories of the studied
processes [7], and, thirdly, to study the how the asymptotic behavior of BRWs dependon the structure of the spectrum of operators that determine the processes of walks andbranching.
54

The study has been carried out at Steklov Mathematical Institute of Russian Academyof Sciences, and was supported by the Russian Science Foundation (RSF), project no. 2311-00375.
References:
1. Yarovaya E. Branching Random Walks in a Heterogeneous Environment Center ofApplied Investigations of the Faculty of Mechanics and Mathematics of the Moscow State
University, Moscow, Russian, 2007, 104 pp.
2. Molchanov S., Yarovaya E. Large deviations for a symmetric branching random walkon a multidimensional lattice. Proc. Steklov Inst. Math. Vol. 282, 2013, pp. 186–201.
3. Yarovaya E. Influence of the Configuration of Particle Generation Sources on the
Behavior of Branching Walks: A Case Study. In: Operator Theory and Harmonic Analysis.
OTHA 2020, Part II — Probability-Analytical Models, Methods and Applications.
Springer Proceedings in Mathematics and Statistics, Springer International Publishing
AG, Switzerland, Cham, Vol. 358, pp. 387–405.
4. Rytova A., Yarovaya E. Heavy-tailed branching random walks on multidimensionallattices. A moment approach. Proceedings of the Royal Society of Edinburgh: Section A
Mathematics, Vol. 151, No 3, 2021, pp. 971–992.
5. Makarova Iu., Balashova D., Molchanov S., Yarovaya E. Branching Random Walks with
Two Types of Particles on Multidimensional Lattices. Mathematics Vol. 10, no. 6:867,
2022, pp. 1–46.
6. Smorodina N., Yarovaya E. Martingale method for studying branching random walks.
Russian Math. Surveys, Vol. 77, No 5, 2022, pp. 955–977.
7. Smorodina N., Yarovaya E. On one limit theorem for branching random walks. Theory
Probab. Appl., 2023 (in print).

Branching Processes with Migration Subordinated byRenewal Process
Yanev George
University of Texas Rio Grande Valley, USA
and Institute of Mathematics and Informatics, Bulgaria
george.yanev@utrgv.edu

Keywords: Controlled branching process; renewal process; regenerative process; random
time change; branching process with migration.
In [1], Nikolay Yanev introduced control branching processes with random control
functions, known as ϕ-branching processes. Available results for this general class ofprocesses, were presented in [2]. Recently, ϕ-branching processes with continuous time
were introduced as well as limit theorems obtained in [3].
Using a renewal process as subordinator, we study branching processes with migrationin continuous time. For these processes, we derive limit theorems assuming the offspringmean is one (critical case) and the emigration prevails over the immigration on average.

55

References:
1. Yanev, N.M. Conditions of extinction of ϕ-branching processes with random ϕ. Theory
Prob. Appl., Vol. 20, 2, 1975, pp.433-440.
2. Gonzalez, M., del Puerto, I. and Yanev, G.P. Controlled Branching Processes, London,
ISTE Ltd. and John Wiley & Sons, 2018, 237.
3. Gonzalez, M., Molina, M., del Puerto, I., Yanev, N.M., and Yanev, G.P. Controlled
Branching Processes with Continuous Time. J. Appl. Prob., Vol. 58, 3, 2021,
pp.830вЂ“848.

Theta positive branching processes in varying
environment
Zhumayev Yerakhmet
L.N. Gumilev Eurasian National University, Astana, Kazakhstan
yerakhmet@gmail.com

Keywords: branching process, varying environment, theta branching, inhomogeneous
Markov process.
This talk is based on the joint work [1] dealing with a time inhomogeneous Markov
branching process {Zt }t≥0 with Z0 = 1. It is a stochastic model for the fluctuating sizeof a population consisting of individuals that live and reproduce independently of eachother, provided that the coexisting individuals are jointly effected by the shared varying
environment in the following way:
- an individual alive at time t dies during the time interval (t, t + δ) with probability
λt δ + o(δ) as δ → 0,
- an individual dying at the time t is instantaneously replaced by k offspring withprobability pt (k), where k = 0 or k ≥ 2.
The time-dependent reproduction law of this model is summarized by two functions
Z tΛt =λu du, ht (s) = pt (0) + pt (2)s2 + pt (3)s3 + . . . ,0where ht (s) is the probability generating function for the offspring number and Λt ,assumed to be finite for all t ≥ 0, is the cumulative hazard function of the life lengthof the initial individual.
t (s)In terms of the mean offspring number at = ∂h∂s|s=1 , also assumed to be finite for allt ≥ 0, the mean population size µt = E(Zt ) has the following expression
Z t
µt = exp(au − 1)dΛu .
0

Putting mt = E(Zt |Zt >0), observe that µt = mt P(Zt > 0). Let q = lim P(Zt =0) as t → ∞be the extinction probability of the branching process.
56Compared to the time homogeneous setting, the added feature of varying environment
makes the model very flexible and therefore cumbersome to study in the most general
setting. In this paper, we distinguish between five classes of the branching processes invariable environment(i) supercritical if q < 1 and lim µt = ∞,(ii) asymptotically degenerate if q < 1 and lim inf µt < ∞,(iii) critical if q = 1 and lim mt = ∞,(iv) strictly subcritical if q = 1 and lim mt ∈ [1, ∞),(v) loosely subcritical if q = 1 and lim mt does not exist.
The division into five classes (i)-(v) is a modified version of the classification suggested
in [2] for the branching processes in variable environment with discrete time. In [2], theclasses (iv) and (v) are considered as one class called subcritical.
Paper [1] focuses on a special family of branching processes in variable environment
which we call theta-positive branching process, cf [3], with the branching parameter θ ∈(0, 1] in varying environment ({λt }, {at }). The branching parameter θ controls the higher
moments of the offspring distribution specified by the formula
ht (s) = 1 − at (1 − s) + at (1 + θ)−1 (1 − s)1+θ .
It is assumed that the fluctuations of the mean offspring number at are restricted to a
fixed interval
0 ≤ at ≤ 1 + 1/θ.
The key feature of the theta-positive branching process Zt is the explicit probability
generating function−θE(sZt ) = 1 − (µ−θ
+ Bt (θ))−1/θ ,t (1 − s)Rtwhere Bt (θ) = θ(1 + θ)−1 0 µ−θ au dΛu .
For the presentation purposes, we consider the important special case of (2) with θ = 1,
when pt (0) = 1 − at /2 and pt (2) = at /2, the theta-positive branching process turns into
the classical birth and death process in varying environment. In this case the generating
function is linear-fractionalE(sZt ) = 1 −µt (1 − s),
1 + µt (1 − s)Bt

Rt
with Bt = 21 0 µ−1
u au dΛu .
Thefollowing
theorems presents our main results in the case θ = 1 in terms of Vt =R
1 t −1µ dΛu , V = lim Vt , Λ = lim Λt .
2 0 u
Theorem 1. If V < ∞, then q < 1, lim µt = µ, 0 < µ ≤ ∞, and q = (V + 12 + 12 µ−1 )−1 .
If V = ∞, then q = 1 and P(Zt > 0) ∼ (Vt + 21 µ−1 )−1 , mt ∼ µt Vt + 21 .
Theorem 2. A theta-positive branching process is supercritical if and only if V < ∞,and Λ = ∞. In this case, lim µt = ∞, q = (V + 12 )−1 and µ−1t Zt almost surely converges
1−wW
to a random variable W such that E(e) = 1 − (V + 2 + 1/w)−1 .
Theorem 3. A theta-positive branching process is asymptotically degenerate if andonly if Λ < ∞. In this case, lim µt = µ, 0 < µ < ∞ and Zt almost surely converges toa random variable Z∞ such that E(sZ∞ ) = 1 − 1/(V + 21 (1 − 1/µ) + 1/(µ(1 − s))).
Corollary 1. If Λ < ∞ and at ≡ 0, then the theta-positive branching is asymptoticallydegenerate with µ = e−Λ and E(sZ∞ ) = 1 − µ + µs.
57

Theorem 4. A theta-positive branching process is critical if and only if V = ∞and µt Vt → ∞. In this case, P(Zt > 0) ∼ 1/Vt , mt ∼ µt Vt and lim E(e−wZt /mt ) =1 − 1/(1 + 1/w), w ≤ 0.
Corollary 2. If Λ = ∞ and 0 < lim inf µt ≤ lim sup µt < ∞, then the theta-positive
branching is critical.
Theorem 5. A theta-positive branching process is strictly subcritical if and only if
V = ∞ and µt Vt → M, 0 ≤ M ≤ ∞. In this case, µt → 0, P(Zt > 0) ∼ mµt , mt ∼m, m = M + 12 and E(sZt |Zt > 0) → 1 − m/(M − 21 + 1/(1 − s)).
Theorem 6. A theta-positive branching process is loosely subcritical if and only if
V = ∞ and µt Vt does not have a limit. In this case, there are several subsequencest0 = {tn } leading to different partial limits µt Vt → M, t0 → ∞.
References:
1. Sagitov S., Lindo A., Zhumayev Y. Theta-positive branching in varying environment.
Preprint (2023) arXiv:2303.04230.
2. Kersting G. A unifying approach to branching processes in a varying environment.
J. Appl. Probab., 57, 2020, pp. 196-220.
3. Sagitov S., Lindo A. A special family of Galton-Watson processes with explosions. In
Branching Processes and Their Applications. Lect. Notes Stat. Proc., Berlin, Springer,
2016, 17 p.

Branching selfdecomposability and limit theorems forsuperposition of point processes
Zuyev Sergei, Molchanov Ilya
Dept. of Mathematical Sciences Chalmers U. of Technology and U. of Gothenburg,
SwedenU. of Bern, Switzerland
sergei.zuyev@chalmers.se, ilya@stat.unibe.ch

Keywords: point process, continuous branching, limit theorems, superposition,
selfdecomposability, stability.
Limit theorems for superposition of independent point processes (PPs) must involvean operation that makes them ”thinner” so that a limit of superposition of their growing
number exist. It is analogous to scaling for random variables, but to preserve a PP
framework, this "scaling"of a PP must be stochastic acting independently on the PP’points. We show that the most general such operation on a PP is independent branchingof its points with diffusion. The simplest example is given by pure-death process without
displacement of points which is equivalent to independent thinning of points. Given suchan operation, one can formulate limit theorems for superposition of independent PPs
aiming to characterise all possible limits. The processes which may arise as a limit are
selfdecomposable (SD) PPs which are a strict subclass of infinitely divisible (ID) PPs. At
the same time, it is strictly larger than the class of strictly stable PPs which arise as alimit of scaled superposition of i.i.d. PPs. Since SD PPs are also ID, their distribution ischaracterised by Levy measure (also known as KLM measure in PP context) and it has
58

a special integral representation from potential theory and the theory of general Markov
processes. We fully characterise the Levy measures of SD PPs and prove their series
decomposition which is a generalisation of LePage series known for stable PPs [1, 2] andwhich mimics the stochastic integral representation of SD random variables.
References:
1. Davydov, Yu. and Molchanov, I. and Zuyev, S. Stability for random measures, point
processes and discrete semigroups. Bernoulli, Vol. 17, No 3, 2011, pp. 1015–1043.
2. Zanella, G. and Zuyev, S. Branching-stable point processes. Electronic J. Probab.,Vol. 20, No 119, 2015, pp. 1–26

59

60

Path 2

Stochastic analysis

61

Estimation of the Parameter of one Class ofDistributions
Adirov T.Kh.
Fiscal Institute under the Tax Committee, Tashkent, Uzbekistane-mail:tolliboyadirov1960@gmail.com

Keywords: Random variables, Order Statistics, Estimating the Parameter ofDistribution.
Let X1 , X2 , ..., Xn be independent, non-negative and identically distributed random
(1)
(2)
(n)
variables (r.v.) with common distribution function (d.f.) F (x). Xn , Xn , ..., Xn are order
(1)
(2)
(n)
statistics of r.v X1 , X2 , ..., Xn , arranged in descending order, i.e. Xn ≥ Xn ≥ ... ≥ Xn .
We introduce the following class of d.f. F (x) defined on (0, ∞):

o
n
 1
Rα = F (x) : 1 − F (x) = exp −x α L(x) , x ≥ x0 > 0 ,where 0 < α < ∞ and
Z x


ε(t)
L(x) = exp
dt , a > 0, ε(t) → 0 as t → ∞.
tαIn what follows, we will assume that parameter α entering the definition of this classis unknown. For estimating parameter α of this class, the author in [1] proposed andstudied estimates involving k extremal order statistics or all elements of order statistics
with weight coefficients. This article is a continuation of studies conducted in [1]; anestimate, which consists of one element of the order statistics for the following parameteris proposed:(k)log Xnαk,n =, for 1 < k < nlog log nk
We assume that the d.f. F (x) is strictly increasing and continuously differentiable. The
main result of this article is reduced to the following theorem.
Theorem. Let k → ∞ as n → ∞ such that k = o(n) and√ n α 
nnk log
log log log → 0kkkThen as n → ∞
√
n
n
dis
k log log log α−1 (αk,n − α + bk,n (α)) =⇒N (0, 1),kk
diswhere symbol =⇒ means convergence by distribution, N (0, 1) are standard Normally
distributed r.v. andlog logα nk
bk,n (α) =→ 0 as n → ∞.
log log k
References
1. Adirov T.Kh. Limit theorems for estimating the parameter of the generalized
Weibull distribution. Abstract of the dissertation of Candidate of Physical andMathematical Sciences, Tashkent, 1993. (in Russian)
2. Seneta E. Regularly Varying Functions. 1976. Springer-Verlag Berlin-HeidelbergNew York.
62

Digits of Powers of 2 in Ternary Numeral System
Aliyev Yagub
ADA University
yaliyev@ada.edu.az

Keywords: Ternary number system, Benford’s law, powers of two.
We study the digits of the powers of 2 in the ternary number system. We propose
an algorithm for doubling numbers in ternary number system. Using this algorithm, we
explain the appearance of blocks of 2s and 0s when the number 2n+1 is written on top
of 2n (n = 0, 1, 2, ...) in a natural way so that for example the last digits are forming
one column, the second to the last digits are forming another column, and so forth. We
also look at the patterns formed by the first digits, the patterns formed by the last digits
and use this to prove that the sizes of these blocks of 0s and 2s are unbounded. We also
discuss how this regularity changes when the digits move from left end of the numbers to
the right end. Let us write first powers of two (1, 2, 4, ..., 215 ) in ternary numeral system
so that their digits in the corresponding place values are aligned along vertical columns.
1 1 2 2 2
2 1 1 1
1 0 2 0
1 2 1
2 2
1 1
2
1

2
1
2
2
1
0
0
0
1

1
0
0
1
0
1
0
0
1
2
1

1
2
1
2
2
2
2
1
2
1
0
1

2
1
0
0
1
2
2
1
0
0
1
2
2
1

2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1

We discuss the patterns occurring in base 3 representation of powers of 2. We showthat
1. every string of ending digits appears infinitely often, provided the string does not
end in 0,
2. every string of starting digits (not beginning with 0) appears infinitely often,
3. if the powers of 2 are all written in base 3 as one column so that the digits of thesame place value are on top of each other, then the size of the triangular blocks ofzeros and twos grow indefinitely.
Part 1 was proved using number theory methods, but it can also be proved using the factthat 2 is a primitive root modulo each power of 3. Part 2 was proved using the fact that63

for irrational number α, the sequence xn = {nα}, where {x} is the fractional part of x, is
uniformly distributed in [0, 1], but it can also be interpreted in the context of "Benford’slaw". Part 3, which is the main objective of the current study, is shown to be a directconsequence of the Parts 1 and 2. Also, the change of the distribution of probabilities ofthese combination of digits when they are taken in between the left and right endpoints,
is studied.
Theorem 1. If the powers of 2 are written so that each next power of 2, in ternary
number system notation, is written on top of the previous power of 2, and the digits
corresponding to the same place values are all on the same vertical lines, then arbitrarily
large triangular blocks of zeros (twos) can appear in this infinite triangular table.
The current study is related to the specific question asked by P. ErdЕ‘s: how frequently
do the powers of 2 have ternary expansions that omit the digit 2? He conjectured that
this holds only for finitely many powers of 2. See [1] for the discussion of this problem.
In view of the results of the current study, ErdЕ‘sвЂTM s conjecture can be interpreted in
the way that there are only finitely many powers of 2 which does not intersect the blocks
containing only twos.
Theorem 2. The probability of an m-digit number A, which may or may not startwith zero digit, appearing at kth position (k > 1) from left of 3-base representations of
2n , is

3k+1
Y−1 
1
.
pk (A) = log3
1+ m
3 i+A
k
i=3

The obtained result generalize the results given in [2,3] about the significant digit
phenomenon also known as The Significant-Digit Law. In particular, this means thata sequence of m zeros 00...0 is more likely to appear towards the left side of the table
than a sequence of m twos 22...2, but as the block of m digits approach the right side of
the construction then the probabilities become more unified.
References:
1. Lagarias, J.C., Ternary expansions of powers of 2. Journal of the London MathematicalSociety, 79, 2009, pp. 562-588.
2. Hill, T., A Statistical Derivation of the Significant-Digit Law, Statistical Science, 10, 4,
1995, pp. 354-363.
3. Hill, T.P., The Significant-Digit Phenomenon, The American Mathematical Monthly,
102, 4, 1995, pp. 322-327.

64

Central limit theorem for stochastic perturbations of
PL circle maps with two break points
Aliyev Abdurakhmon, Tirkasheva Gulasal
Institute of Mathematics of the Academy of Sciences of Uzbekistan
National University of Uzbekistan named after Mirzo Ulugbek
aliyev95.uz@mail.ru

Keywords: Central limit theorem, circle dynamics, stochastic perturbation, markov chain.
One of the classical problems of the theory dynamical systems is asymptotic behaviorof stochastic perturbations of one-dimensional dynamics. J. Crutchfield et al. [1] andB. Shraiman et al. [2] heuristically considered in their works a renormalization grouprespectively a field theoretic path-integral approach for weak Gaussian noise perturbing
one dimensional maps with period doubling at the onset of chaos. E.Vul, Y.Sinai, andK.Khanin [3] studied the effect of noise on the ergodic properties of these maps andshowed that for systems with weak noise at the accumulation of period doubling, there
is a stationary measure, depending on the magnitude of the noise, which converges
for vanishing noise to the invariant measure of the attractor. O.Espinosa and R.Llave[4] studied stochastic perturbations of several systems using the renormalization group
technique and proved a central limit theorem for critical circle maps with a golden meanrotation number and some mild conditions on the stochastic noise. A. Dzhalilov, D. Mayer
and A. Aliyev [5] investigated this problem for circle maps with a break point. We extend
these results for piecewise linear (PL) circle maps with bounded type irrational rotation
number.
Now, we turn to the formulation of the main results of our work.
Let (Ω, F, P ) be a probability space and T be a PL circle homeomorphism. Let thestochastic sequence defined asx̄n+1 = T (x̄n ) + σξn+1 , x̄0 := x ∈ S 1
where (ξn ) be a sequence of independent random variables with p > 2 finite moments
satisfying following conditions:Eξn = 0(1)
1

1

const ≤ (E|ξn |2 ) 2 ≤ (E|ξ|p ) p ≤ Const.
Let ωn (x, σ) be the stochastic process defined byx̄n − xn
ω(x, σn ) = p
σn var(x̄n )

(2)

(3)

We formulate the main result of this work.
Theorem. Let T be a piecewise linear circle homeomorphism with bounded typeirrational rotation number and two break ponts b1 and b2 on different orbits. Consider
a sequence of independent random variables (ξn ) with p > 2 finite moments satisfying theconditions (1) and (2). Then there exist a constant γ > 1 depending on T and p, for all
sequences σn satisfying the condition:
lim σn nγ = 0,n→∞

the process ωqn (x, σqn ) converges in distribution to the standard Normal random variable.
65References:
1. J.Crutchfield, M.Nauenberg, and J.Rudnick. Scaling for external noise at the onset ofchaos. Physical Review Letters, 46(14):933-935, 1981.
2. B.Shraiman, C.E.Wayne, and P.C.Martin. Scaling theory for noisy period-doublingtransitions to chaos. Physical Review Letters, 46(14):935-939, 1981.
3. E.B.Vul, Ya.G. Sinai, and K. M. Khanin. Feigenbaum universality and thethermodynamic formalism. Russian Math. Surveys, 39(3):1-40, 1984.
4. O.Diaz-Espinosa and R.de la Llave. Renormalization and central limit theorem for
critical dynamical systems with weak external noise.JMD 1(3) 477-543 2007
5. A. Dzhalilov, D. Mayer, A. Aliyev. The thermodynamic formalism and the central
limit theorem for stochastic perturbations of circle maps with a break. Russian Journalof Nonlinear Dynamics, 2022, vol. 18, no. 2, pp. 253-287.

SIS model on (random) dense graphs: probabilistic
approach and remarks on optimal vaccinations
Delmas, Jean-François
Ecole des Ponts, France
jean-francois.delmas@enpc.fr

Keywords: heterogeneous SIS model, individual based model, reproduction number,vaccination, herd immunity, equilibrium.
We consider an individual based model on a random dense graph for a SIS epidemic
in an heterogeneous population of size n :- Each individual i has a feature (age, localization, ...), say xi ∈ X.
(n)

- Individuals i and j are connected with probability wE (xi , xj ) ∈ [0, 1].
- Individual i is either Susceptible (S) or Infectious (I), and denote by Eti its state at timet ≥ 0.
(n)- Individual j infects individual i at rate wI (xi , xj ) ≥ 0 (provided that i is S, j is I andi, j are connected).
- Individual i recover at rate γ(xi ) > 0 (provided that i is I).
The infected population at time t is described by the random measure:n(n)ρt (dx) =1X
1 t δx (dx).
n i=1 Ei =I i(n)Under suitable conditions (see [1]), the random measure ρt converges to a deterministic
limit ρt (dx) = u(t, x)µ(dx), with u(t, x), the infected fraction at time t of individuals withfeature x, unique solution to:Z∂t u(t, x) = (1 − u(t, x))k(x, x0 )u(t, x0 )µ(dx0 ) − γ(x)u(t, x),
x0 ∈X

66

where µ(dx0 ) is the (asymptotic) distribution of the feature x0 in in the population and:
(n)

(n)

k(x, x0 ) = lim nwE (x, x0 )wI (x, x0 ).
n→∞

The proof use a coupling argument with a SIS model on a complete graph.
The reproduction number R0 is then the spectral radius of the integral operator withkernel k(x, x0 )/γ(x0 ), see [2]. If R0 ≥ 1, 0 is the only equilibrium of the ODE andlim u(t, ·) = 0; if R0 > 1 then there exists a non-zero maximal equilibrium, say g ∗ .
t→∞
The use of a perfect vaccine at time t = 0, can be interpreted as replacing the population
distribution µ(dx) by the effective population distribution η(x) µ(dx), where η(x) is thefraction of un-vaccinated population with feature x, see [2,3]. The corresponding effective
reproduction number Re (η) is the spectral radius of the integral operator with kernelk(x, x0 )η(x0 )/γ(x0 ). In particular, vaccinating uniformly the population with probability
1 − 1/R0 (that is η unif = 1/R0 ) is critical as Re (η unif ) = 1 (so that the epidemic vanishes
asymptotically). We give a rigorous proof in [4] of the intuitive fact that vaccinating afraction g ∗ (x) of the population with feature x, that is η = 1 − g ∗ , is critical:
Re (1 − g ∗ ) = 1.
This is joint work with D. Dronnier, P. Frasca, F. Garin, V. C. Tran, A. Velleret and P.-A.
Zitt.
References:
1. J.-F. Delmas, P. Frasca, F. Garin, V. C. Tran, A. Velleret, and P.-A. Zitt. Individual
based SIS models on (not so) dense large random networks. Arxiv (2023)
2. J.-F. Delmas, D. Dronnier, and P.-A. Zitt. An infinite-dimensional metapopulation SIS
model. Journal of Differential Equations (2022).
3. J.-F. Delmas, D. Dronnier, and P.-A. Zitt. Targeted vaccination strategies for an infinitedimensional SIS model. ArXiv (2022).
4. J.-F. Delmas, D. Dronnier, and P.-A. Zitt. Vaccinating according to the maximal
endemic equilibrium achieves herd immunity. ArXiv (2022).

Note on critical mappings of the circle with rotation
number of algebraic type
Akhtam A.Dzhalilov, Saidakhmat Kh.Abdukhakimov.
Turin Polytechnic University in Tashkent,Uzbekistan. Tashkent. Uzbekistan.
Jizzakh branch of National University of Uzbekistan. Jizzakh. Uzbekistan.
a-dzhalilov@yahoo.com, asaidahmat@mail.ru

Keywords: circle homeomorphisms, rotation number, hiting time.
Consider the space Xcr (ρ) of real-analytic homeomorphisms of the circle with rotation
number ρ = [k1 , k2 , k1 , k2 , ...] , k1 , k2 ∈ N. And with one critical point at which thederivative is early zero. It is well known (see [1]) that the renormalization transformation
Rcr = Rcr (k1 , k2 ) into Xcr (ρ) has a single fixed point Tcr = Tcr (k1 , k2 ). Denote by Cr (Tcr )the set of all critical C 1 -conjugate to Tcr maps of the circle.In the work (see [2], [3]) a
67

unique pair (Uk1 , Uk2 ) of potentials was constructed corresponding to all mappings Xcr (ρ)
from.The main goal of this work is the behavior of the normalized hit times in smallneighborhoods of the critical point.
(1)(1)
The distribution function Φn (t) and the corresponding first hit function En (t) canbe expressed using the lengths of segments of dynamic partition elements Pn .
We now formulate the main result of our work.
Theorem. Consider
theo critical mapping T ∈ Cr (Tcr ) . And the sequence ofn(1)

distribution functions Φn (t)(1)∞

corresponding to the normalized function of the firstn=1

hit E n (x) in the n−th renormalization neighborhood of the singular point x0 .
Then1) For all t ∈ R1 there are finite limits and the following relations are true:
(1)

(2)

(1)

(2)

limn→∞ Φ2n−1 (t) = Φk1 (t) , limn→∞ Φ2n (t) = Φk2 (t) ;
(2)

(2)

(2)

(2)

2) Φk1 (t) = 0, Φk2 (t) = 0, if t ≤ 0 and Φk1 (t) = 1, Φk2 (t) = 1, if t ≥ 1(2)(2)3) Both functions Φk1 (t) and Φk2 (t) are continuous on the line.
References:
1. Ostlund R., Rand D., Sethna J., Sigga E. Physica D. 1983. V. 8.303.
2. Abdukhakimov S.X., Khomidov M.K. The orbit of critical point and thermodynamic
formalism for critical circle maps without periodic points. Uzbek Mathematical Journal,
2020 № 3pp. 4-15.
3. Dzhalilov A.A. Limit hitting time laws for critical circle mappings, TMF, 2004,
volume 138, number 2, 225-245.

Convergence rate estimates in the Hartman-Wintner
Law of the Iterated Logarithm
Gafurov M. U.
Tashkent State Transport University
mgafurov@rambler.ru

Keywords: law of the iterated logarithm, convergence of a series, number of exits, normallaw, rate of convergence in the central limit theorem
The work is devoted to further refinement of the classical Hartman-Wintner theoremon the law of the iterated logarithm for a sequence of independent, identically distributed
random variables. Namely, an estimate of the rate of convergence in the form of convergent
series of weighted probabilities of large deviations is established – the exact asymptotes
in the small parameter of the series, which is a refinement of the corresponding result[1]. Analogs of the obtained results were proved for a family of independent, identically
distributed random variables indexed on sectors of the d-dimensional lattice of the
Euclidean space
References:
1. Gafurov M. U. On the estimate of the rate of convergence in the law of iterated
logarithm. in: Probability Theory and Mathematical statistics (Tbilisi, 1982), in: Lecture
Notes in Math., Vol. 1021, 1983, p. 137-144.
68

Quadratic Stochastic Operators with Countable StateSpace
Ganikhodjaev, N.N.
Institute of Mathematics of the Academy of Sciences of Uzbekistan
nasirganikhodjaev@gmail.com

Keywords: regular transformation, Volterra quadratic stochastic operator, infinite state
space.
Let (X, F) be a measurable space, where X is a state space and F is σ-algebra on
X, and S(X, F) be the set of all probability measures on (X, F). Let {P (x, y, A) : x, y ∈X, A ∈ F} be a family of functions on X × X × F such that P (x, y, ·) ∈ S(X, F), whereP (x, y, A) be regarded as a function of two variables x and y with fixed A ∈ F, is ameasurable function on (X × X, F ⊗ F) and P (x, y, A) = P (y, x, A) for any x, y ∈ X andA ∈ F.
Specifying such family of functions {P (x, y, A) : x, y ∈ X, A ∈ F} we introduce a nonlinear transformation V : S(X, F) → S(X, F) which is defined by
Z Z
P (x, y, A)dλ(x)dλ(y),(V λ)(A) =X

X

where A ∈ F is an arbitrary measurable set.
The case with finite state space X was considered by Bernstein [1].
When state space X is an infinite countable set of positive integers, and F is the power
set P(N ) of N, i.e. the set of all subsets of N, thenSN

= {x = (xi )∞i=1 : ∀ixi ≥ 0,∞X

xi = 1}i=1

is the set of all probability measures on (N, P(N )).
In this case, the measure P (i, j, ·) is a discrete probability measure and the corresponding
qso V : S N → S N has the following form(V x)k =∞X

Pij,k xi xj , k ∈ N

(1)

i,j=1

where the coefficients Pij,k satisfy the following conditions:a)Pij,k ≥ 0; b)Pij,k = Pji,k ; c)∞X

Pij,k = 1 for all i, j, k ∈ N.

k=1

The quadratic stochastic operator V (1) is called Volterra, if pij,k = 0 for any k ∈/ {i, j}.
The biological treatment of such operators is rather clear: the offspring repeats one of its
parents.
A qso V is a Volterra if and only if(V x)k = xk (1 +

∞
X
i=1

69

aki xi )where A = (aij )∞1 is a skew-symmetric matrix with aki = 2pik,k − 1 for i 6= k, aii = 0 and|aij | ≤ 1. Here i, j ∈ N.
A qso V is called a regular if for any initial point x ∈ S N , a limit
lim V n (x)n→∞exists.
Note that the limit point is a fixed point of a qso V. Thus the fixed points of qso describe
a limit or a long run behavior the trajectories at any initial point. A limit behavior of
trajectories and the fixed points of qso play an important role in many applied problems[2],[3].
A nonlinear operator V defined on the finite-dimensional simplex S m−1 is called ergodic
if the limit
n−1
1X k
lim
V (x)
n→∞ n
k=0
exists for any x ∈ S m−1 . It is evident that a regular qso V is ergodic; however, regularity
does not follow from the ergodicity
On the basis of numerical calculations, Ulam conjectured [4] that an ergodic theorem
holds for any qso V. In 1977, Zakharevich [5] proved that in general this conjecture isfalse. He considered the following Volterra operator on S 2
x01 = x1 (1 + x2 − x3 )x02 = x2 (1 − x1 + x3 )x03 = x3 (1 + x1 − x2 )and proved that it is a non-ergodic transformation.
In this paper we consider the limit behaviour of the trajectories of the following Volterra
operator with infinite state space as follows.
(V x)1 = x1 [1 + ax2 − ax3 + ax4 − ax5 + ax6 − ax7 + · · · ](V x)2 = x2 [1 − ax1 + ax3 − ax4 + ax5 − ax6 + ax7 − · · · ]······(V x)2n−1 = x2n−1 [1 + ax1 − ax2 + · · · − ax2n−2 + ax2n − ax2n+1 + ax2n+2 − ax2n+3 + · · · ](V x)2n = x2n [1 − ax1 + ax2 − · · · − ax2n−1 + ax2n+1 − ax2n+2 + · · · ]······where a ∈ [−1, 1].
References:
1. Bernstein S.N. The solution of a mathematical problem related to the theory ofheredity.Uchn. Zapiski. NI Kaf. Ukr. Otd. Mat., Vol.1 , No. 1 , 1924, pp. 83-115 .
2. Kesten, H. Quadratic transformations: A model for population growth.
IAdv.Appl.Prob.,, Vol.2 , 1970, pp. 1-82 .
3. Lyubich Yu.I.Mathematical structures in population genetics. Biomathematics 22,
70

Springer-Verlag,1992.
4. Ulam, S.A collection of mathematical problems,Interscience Publishers,New-YorkLondon 1960.
5.Zakharevich M.I. On behavior of trajectories and the ergodic hypothesis for quadratictransformations of the simplex,Russian Math.Surveys Vol.33,1978, 265-266.

Long-term behaviour of the neutron transport equation
at criticality
Emma Horton
University of Warwick
emma.horton@warwick.ac.uk

Keywords: neutron transport, criticality, asymptotic moments, Perron-Frobenius, Yaglom,
survival probability
The neutron transport equation (NTE) describes the flux of neutrons over time throughan inhomogeneous fissile medium. Understanding the long-term behaviour of solutions
to the NTE is vital for nuclear safety and reactor design. One way to do this is toconsider the so called k-effective eigenvalue problem. The eigenvalue, keff , has the physicalinterpretation as being the ratio of neutrons produced (during fission events) to thenumber lost (due to absorption in the reactor or leakage at the boundary) per typicalfission event and determines the criticality of the system. In this talk, we will prove theexistence of keff and the corresponding eigenfunctions via a Perron Frobenius type result
by modelling the nuclear fission process as an appropriate discrete time branching process.
We will then focus on the critical case (keff = 1) and discuss the limiting behaviour of theprocess.
This talk is based on joint work with Alex M. G. Cox, Eric Dumonteil, Andreas
Kyprianou, Denis Villemonais and Andrea Zoia.

Weakly Dependent Properties of Vertex Processes of aConvex Hull
Khamdamov Isakjan
National University of Uzbekistan named after Mirzo Ulugbek
and University of Public Safety of the Republic of Uzbekistan, Tashkent, Uzbekistane-mail: khamdamov.isakjan@gmail.com

Keywords: Convex Hull, Binomial Point Process, Poisson Point Process, Vertex
Processes.
The work is devoted to the study of the properties of convex hulls generated by the
implementation of a inhomogeneous Poisson point process on the unit disk.
71

Efron B. [1], Reny A. and Sulanke R. [2] and other researchers were the first to study
the functionals of the convex hull; they found the asymptotic behavior and revealed theconnections between the asymptotic expressions for the mathematical expectations of thenumber of vertices, area and perimeter of the convex hull in the case when random points
are uniformly distributed in the square. Carnal H. in [3], obtained asymptotic expressions
for similar convex hull functionals generated by random points set in polar coordinates.
Their components are independent of each other, the angular coordinate is uniformly
distributed, and the tail of the distribution of the radial coordinate is a regularly varying
function near the boundary of the support - the disk or at infinity.
Using the approximation of a binomial point process to a homogeneous Poisson process,
Groeneboom P. in [4], managed to prove the central limit theorem for the number ofvertices of a convex hull, for the case when the support of the original uniform distribution
is a convex polygon or ellipse.
In this article, the property of strong mixing and the martingale property of functionalsof vertex processes of the convex hull are established, in the case when the convex hull isgenerated from a inhomogeneous Poisson point process inside the disk.
References
1. Efron B. The convex hull of a random set of points, Biometrika, Vol.52, 1965,
pp.331–343.
2. Reny A. and Sulanke R. Uber die konvexe Hulle von n zufalling gewahlten Punkten,
Z. Wahrscheinlichkeitstheorie verw. Geb., Vol.2, 1963 pp.75–84.
3. Carnal H. Die konvexe Hulle von n rotationssymmetrisch verteilten Punkten, Z.
Wahrscheinlichkeitstheorie verw. Geb., Vol.15, 1970, pp.168–176.
4. Groeneboom P. Limit theorems for convex hull, Probab. Th. Rel. Fields, Vol.79,
No.3, 1988, pp.327–368.

On the distribution of the crossing number of a strip bytrajectories of the Levy process
V.R. Khodjibayev1 , V.I. Lotov2
1
Namangan Engineering-construction Institute, Namangan,Uzbekistan,
Institute of Mathematics, Uzbekistan Academy of Scienses,
2
Sobolev Institute of Mathematics, Novosibirsk, Russia
e-mail: vkhodjibayev@mail.ru, lotov@math.nsc.ru
Keywords: stochastic Levy process, crossing number of a strip, asymptotic formulas,
probabilistic inequalities
Let ξ(t), t ≥ 0, ξ(0) = 0, be a Levy process, i.e., a homogeneous stochastic processwith independent increments whose sample functions are continuous on the right. For it
we have
E exp{λξ(t)} = exp{tψ(λ)},σ 2 λ2ψ(λ) = γλ +
+
2

Z∞ 
eλx − 1 −

−∞

72


λx
dS(x), (1)1 + x2

with standard conditions on γ, σ and S(x). Introduce two sequences of stopping times:
τ0+ = τ0− = 0,+
τi− = inf {t > τi−1
: ξ(t) ≤ −a},τi+ = inf {t > τi− : ξ(t) ≥ b},i ≥ 1,

where a > 0, b > 0. Define a random variable θ equal to the number of upcrossings ofthe strip {−a < y < b} on the plane (x, y) by trajectories of the process (t, ξ(t))∞
t=0 . Thisvariable is finite with probability one if Eξ(1) 6= 0.
Our goal is to study the distribution of θ. The exact calculation of it is available only insome particular situations. The talk is devoted to finding asymptotic formulas for P(θ ≥ t)
as b → ∞ and estimates for this probability.
It is easily seen thatP(θ ≥ k) = P(τk+ < ∞),k ≥ 1.

I. Asymptotic formulas.
Suppose that Eξ(1) < 0 and put, for Re l = 0, x ≥ 0, and y ≤ 0,
ζ = sup ξ(t),Q(x) = P(ζ ≥ x),t≥0

η− (y) = inf{t ≥ 0 : ξ(t) ≤ y},

χ− (y) = ξ(η− (y))−y,P(χ− < t) = lim P(χ− (y) < t).
y→−∞Asymptotic behavior of P(θ ≥ k) is based on the asymptotic properties of ζ and χ− . Letρ = sup{λ ≥ 0 : E exp{λξ(1)} ≤ 1} = sup{λ ≥ 0 : ψ(λ) ≤ 0}.
The following result is known ([1]).
Lemma 1. Suppose that ρ > 0,

ψ 0 (ρ) = E ξ(1) eρξ(1) < ∞. Thenψ(ρ) = 0,

Q(x) = c e−ρx (1 + o(x)),

x → ∞,

wherec−1 = ρψ 0 (ρ)

Z0eρy dF− (y),Z∞
F− (y) = −−∞P( inf ξ(s) ≥ y) dt,0≤s≤ty ≤ 0.

0

Theorem 1. Under conditions of lemma 1 we haveP(θ ≥ k) = P(θ ≥ 1)(hc)k−1 e−ρ(k−1)(a+b) (1 + o(1)),

b → ∞,

for all k ≥ 2 and arbitrary a > 0, h = E eρχ− . If, in addition, a → ∞ then for all k ≥ 1
P(θ ≥ k) = (hc)k e−ρk(a+b) (1 + o(1)).
II. Inequalities.
The estimates of this section can be considered as a natural addition to the asymptotic
results. Similar inequalities for random walks generated by sums of i.i.d. random variables
can be found in [2].
The following inequalities hold for all k ≥ 1.
Theorem 2. Let E ξ(1) < 0. Then

73

(1)

P(θ ≥ k) ≤ Qk (a + b).

(2)

If

−r
R

d S(x) = 0 for some r ≥ 0

then

P(θ ≥ k) ≥ Qk (a + b + r).

−∞

Theorem 3. Suppose that E ξ(1) < 0, E |ξ 3 (1)| < ∞ and the function Q(x) is convex
for x > 0. ThenP(θ ≥ k) ≥ Qk (a + b + l),Z∞
where l = 3a3 /a2 ,ai =|xi | d S(x),i = 2, 3.

−∞Theorem 4. Suppose that E ξ(1) < 0, E |ξ 3 (1)| < ∞ and E exp{ρξ(1)} = 1 for someρ > 0. ThenP(θ ≥ k) ≥ s−1 e−ρk(a+b+l) ,
where s = sup0<x<M E eρ(ξ(1)−x) |ξ(1) ≥ x , M = inf{x > 0 : P(ξ(1) ≤ x) = 1}.
Theorem 5. Suppose that E ξ(1) < 0, E |ξ 3 (1)| < ∞ and E exp{λξ(1)} = ∞ forall λ > 0. Consider a process ξ1 (t) which is obtained by replacing jumps of the process
ξ(t) that exceed a number q > 0, by jumps of size q. Thus, the spectral function of therepresentation (1) for the process ξ1 (t) equals(
S(x),
if x ≤ q,
S1 (x) =S(q),if x > q,
and the process ξ1 (t) satisfies conditions of the theorem 4. Then
−ρ1 k(a+b+l)P(θ ≥ k) ≥ s−1,
1 e

where s1 and ρ1 are defined as above by using ξ1 (t) instead of ξ(t), the number l is definedin the theorem 3.
References
1. N. S. Bratijchuk, D. V. Gusak. Granichnye zadachi dlja processov s nezavisimymi
prirashhenijami, Kiev, Naukova dumka, 1990, 263 p. (in Russian).
2. V. I. Lotov, A. P. Lvov. Bounds for the number of crossings of a strip by random
walk paths, Journal of Math. Sciences, Vol. 230, No 1, 2018, pp. 112–117.

74

Boundedness of the classical operators in weighted
quasi-Banach spaces of entire functions1
Korablina Yuliya
Russia, Rostov-on-Don, SFedU; Vladikavkaz, SMI VSC, RAS
anaconda210150@mail.ru

Keywords: weighted spaces of holomorphic functions, weighted composition operator,
Volterra operator, Bergman spaces, Fock spaces.
We consider a problem of boundedness of classical operators acting on weighted quasiBanach spaces of holomorphic functions Hv (G). This space is defined as follows:|f (z)|< ∞}.
z∈G v(z)Hv (G) = {f ∈ H(G), kf kv = supHere G is a domain in the complex plane, H(G) is the space of all holomorphic functions
in G, and v is a weight on G.
Here and bellow X ,→ H(G) is a quasi-Banach space endowed with the quasi-norm
k · k. X ∗ is a dual space of X consisting of all linear continuous functionals on X endowedwith the norm k · k∗ . δz is δ-function for a fixed point z ∈ G.
Theorem 1. Let v be an arbitrary weight on G. Linear operator T : X 7→ Hv (G) isbounded if and only ifa) δz (T ) ∈ X ∗ for all z ∈ G;kδz (T )k∗
< ∞.
v(z)
z∈G

b) supThis result allowes to establish some criteria of the boundedness of weighted composition
and Volterra operators an abstract quasi-Banach space in terms of δ-functions norms. As
a consequence we obtain criteria of the boundedness of the above mentioned operators on
generalized Bergman, Hardy and Fock spaces. In particular cases it is possible to statethese criteria in terms of weights defining spaces and functions giving the composition
operator. In comparison with the previous results (see [1]) we essentially extend the classof weighted holomorphic spaces in the unit disc that admits a realization of Zorboska’s
method. In addition, we develop an extension of this approach to weighted spaces of entire
functions. In this relation we introduce the class of almost harmonic weights and obtainsome estimates of δ-functions norms in spaces dual to the generalized Fock spaces giving
by almost harmonic weights. Finally, these results are applied to classical growth spaces
and the Cesáro operator acting on them.
References:
1. Abanin A.V., Pham Trong Tien. Differentiation and integration operators on weighted
spaces of holomorphic functions. Math. Nachr. – 2017. – Vol. 290, No 8–9. – P. 1144–1162.
2. Abanin A.V., Korablina Yu.V. Boundedness of classical operators in weighted spaces ofholomorphic functions. Vladikavkaz Math. J. – 2020. – Vol. 22, No 3. – P. 5–17.
3. Zorboska N. Intrinsic operators from holomorphic function spaces to growth spaces.
Integr. Equ. Oper. Theory.–2017.–V. 87, No 4.–P. 581–600.
1 The work was carried out with the financial support of the grant of the President of the Russian Federation for young
scientists-candidates of sciences, project МК-160.2022.1.1

75

Stochastic longitudinal oscillations viscoelastic ropewith moving boundaries, taking into account damping
forcesVladislav L. Litvinov and Kristina V. Litvinova
Moscow State University, Moscow, Russian Federation
Samara State Technical University, Samara, Russian Federation
vladlitvinov@rambler.ru, kristinalitvinova900@rambler.ru

Keywords: linear mathematical model, vibrations of a rope, moving boundaries,
stochastic vibrations
At present, reliability issues in the design of machines and mechanisms require more andmore complete consideration of the dynamic phenomena that take place in the designed
objects. The widespread use in technology of mechanical objects with moving boundaries
necessitates the development of methods for their calculation. The problem of oscillationsof systems with moving boundaries is related to obtaining solutions to integro-differential
and partial differential equations in time-variable domains [1-10]. Such tasks are currently
not well understood. Their peculiarity is the difficulty in using the known methods ofmathematical physics, suitable for problems with fixed boundaries. The complexity of thesolutions obtained is explained by the fact that up to now there has not been a sufficiently
general approach to the analysis of the features of the dynamics of such systems. In
connection with the danger of resonance, the study of forced oscillations is of great
importance here. Attempts to investigate this process have been made, but the results
obtained are limited mainly by a qualitative description of dynamic phenomena [1-4]. In
addition, it is recognized that deterministic modeling of systems cannot be adequate forsome types of problems, so it is necessary to switch to probabilistic-statistical, where there
are random variables, stochastic fluctuations. When solving here, mainly approximate
methods are used [5-9], since obtaining exact solutions is possible only in the simplest
cases [10].
If the damping of transverse vibrations is mainly due to the action of external
damping forces, then in the case of longitudinal vibrations, the damping is mainly affectedby elastic imperfections in the material of the vibrating object [5-10]. The study ofviscoelasticity includes the analysis of the stochastic stability of stochastic viscoelastic
systems, their reliability, etc. The paper considers stochastic linear longitudinal oscillationsof a viscoelastic beam with moving boundaries, taking into account the influence of
damping forces. The case of a difference kernel makes it possible to reduce the problemof analyzing a system of stochastic integro-differential equations to the study of a systemof stochastic differential equations. To estimate the expansion coefficients, it is proposed
to apply the statistical numerical Monte Carlo method [11].
References:
1. Savin G.N., Goroshko O.A. Dynamics of a variable length thread Nauk. dumka, Kiev,
1962, 332 p.
2. Samarin Yu.P. On a nonlinear problem for the wave equation in one-dimensional space
Applied Mathematics and Mechanics, 1964. T. 26, V. 3. P. 77-80.
3. Vesnitsky A.I. Waves in systems with moving boundaries and loads Fizmatlit, Moscow,76

2001, 320 p.
4. Lezhneva A.A. Bending vibrations of a beam of variable length Izv. Academy of Sciencesof the USSR. Rigid Body Mechanics. 1970. No. 1. P. 159-161.
5. Litvinov V.L. Solution of boundary value problems with moving boundaries using
an approximate method for constructing solutions of integro-differential equations Tr.
Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences.
2020.Vol. 26, No. 2. P. 188-199.
6. Anisimov V.N., Litvinov V.L. Mathematical models of longitudinal-transverse
vibrations of objects with moving boundaries Vestn. Himself. tech. un-t. Ser. Phys andmat. science, 2015. Vol. 19, No. 2. P. 382-397.
7. Anisimov V.N., Litvinov V.L. Mathematical modeling and study of the resonanceproperties of mechanical objects with a changing boundary: monograph V. L. Litvinov,
V. N. Anisimov - Samara: Samar. state tech. un - t, 2020. 100 p.
8. Litvinov V.L., Anisimov V.N. Application of the Kantorovich - Galerkin method forsolving boundary value problems with conditions on moving boundaries Bulletin of theRussian Academy of Sciences. Rigid Body Mechanics. 2018. No. 2. P. 70-77.
9. Litvinov V.L., Anisimov V.N. Transverse vibrations of a rope moving in a longitudinal
direction Bulletin of the Samara Scientific Center of the Russian Academy of Sciences.
2017. T. 19. No. 4. - P.161-165.
10. Litvinov V.L., Anisimov V.N. Mathematical modeling and research of oscillations of
one dimensional mechanical systems with moving boundaries: monograph V. L. Litvinov,
V. N. Anisimov Samara: Samar. state tech. un-t, 2017. 149 p.
11. Elepov B. S., Kronberg A. A., Mikhailov G. A. and Sabelfeld K. K. Solution ofboundary value problems by the Monte Carlo method. Novosibirsk: Nauka, 1980. 174 p.

Limited Theorem for Stochastic Integrals overSemi-Martingales
Mamatov Kh. M.
Public Safety of the Republic of Uzbekistan, Tashkent, Uzbekistane-mail:khmmamatov@gmail.com

Keywords: Martingale, Semi-martingale, Wiener process, Kolmogorov distance.
Let a sequence of semi-martingales X n = (Xsn , =ns ), X0n = 0, n ≥ 1 be given on someprobability space (Ω, =, P ) with selected filtration flows F n = (=ns ) and satisfying theordinary conditions.
Let, further, W = (Ws , =s ) be a standard Wiener process (with respect to some flow
F = (=s ), s ≥ 0 ).
The main purpose of this article is to give an estimate for the Kolmogorovnndistance
Rn = supx∈RR |F (x) − F (x)|R between distribution  functions F (x) =1
1
P 0 f (s, Xsn )dXsn ≤ x and F (x) = P 0 f (s, W (s))dW (s) ≤ x .
The estimate obtained in this article generalizes the results previously obtainednby the author in [1] and [2] for the Kolmogorov
R distance Rn = sup x∈R |G (x) −1
G(x)| between distribution functions Gn (x) = P 0 f (s, Msn− )dMsn− ≤ x and G(x) =
77

R

1
P 0 f (s, W (s))dW (s) ≤ x , where M n = (Msn , =ns ) are square integrable martingales,
and W = (Ws , =s ) are standard Wiener processes.
References
1. Mamatov Kh. M. A note on the Levi-Prokhorov estimate between distributionsof processes generated by stochastic integrals. Collection of abstracts of the conference:
Theoretical foundations and applied problems of modern mathematics. II part, 2022,
pp.119–120.(in Russian)
2. Mamatov Kh. M. and Grome I. G. On the rate of convergence in the limiting
theorem for stochastic integrals over martingales. Uspekhi Mat. Nauk. Vol.43, Issue. 2,
1988 pp.143–144.(in Russian)

Limited Theorem for Members of the Variational Series
with a Random Sample Size
Mamurov I. N.
Tashkent Financial Institute, A.Temur Street 60A, 100000, Tashkent, Uzbekistan.
e-mail:imamurov58@gmail.com

Keywords: Random sample, Variational Series, Dependent Scheme, Independent
Scheme.
Many publications are devoted to the study of asymptotic distributions of membersof a variational series formed from independent identically distributed random variables
with deterministic sample size.
In this article, we study the asymptotic distributions of the members of the variational
series in the case when the sample size itself is a random variable, i.e. the characteristicsof the general population under consideration are observed (due to certain circumstances)in a random number of trials. Random sample size appears in statistical problems of
reliability theory, queuing theory, sequential analysis, etc. [1]. In this case, the sample
size as a random variable may turn out to be independent on the observed quantities
themselves (let us call this case "independent scheme"), and in some cases dependent onthem (let us call this case "dependent scheme").
The results obtained present general studies of the limiting distributions of the membersof the variational series with a random sample size in the "independent scheme"and"dependent scheme".
Proofs and detailed discussions of the results obtained on the asymptotic distributionsof the members of the variational series with a random sample size are given in [2] and[3].
References
1. Kruglov V.M. and Korolev V.Yu. Limit theorems for random sums.// M. MGU,
1990. 269p.(in Russian)
2. Dzhamirzaev A.A. and Mamurov I.N. Transfer theorems. Monograph. Tashkent,
2019.(in Russian)78

3. Mamurov I.N. Asymptotic distribution of the central variation members in the caseof random sampling volume. Turkish Online Journal of Qualitative Inquiry. V.12, Issue
7, July 2021:4626–4634.

79

Asymptotic approximation associated with generalized
random allocation schemes
Mirakhmedov Sherzod
Institute of Mathematics, Academy of Sciences. Tashkent/ Uzbekistan
shmirakhmedov@yahoo.com

Keywords: Asymptotic expansion, urn models, random allocation, Poisson distribution,binomial distribution..
Urn models (also known as a random allocation schemes) are a useful tool whichallows to formulate and better understand many combinatorial problems in probability
and statistics, see for instane, [1] and [2]. Most general definition of urn models has
been introduced by Mirakhmedov et al [3], where the asymptotic theory and higherorder expansions for the statistic of the form of a sum of functions of frequencies arepresented. Although their work covers many specific urn models it does not cover, suchprobabilistic models as, for instance, random allocation of particles into infinitely manycells, random allocation of particles in sets, and the statistics based on several samples
from a population(s). The generalized random allocation scheme we are interested here is
as follows.
Let ξl (nl ) = (ξl,1 (nl ), ξl,2 (nl ), ...), l = 1, ..., s, be a collection of sequences (random
vectors) of independent non-negative integer random variables (r.v.s) , distribution of
ξl,m (nl ) depend on parameter nl ∈ ℵ = {1, 2, ...} and such that for each nl the series
ζl (nl ) = ξl,1 (nl ) + ξl,2 (nl ) + .... is a.s. converges, moreover this parameter nl is such thatP r{ζl (nl ) = nl } > 0 ,l = 1, ..., s . Further, let ηl (nl ) = (ηl,1 (nl ), ηl,2 (nl ), ...) be a r.vec.,
distribution of which can be represented as the joint conditional distribution of ξl (nl )given ζl (nl ) = nl , viz.,=(ηl (nl )) = =(ξl (nl ) | ζl (nl ) = nl ), l = 1, ..., s,where =(X) stands for the distribution of the r.vec. X. This equality implies that
ηl,m (nl ) ≥ 0 , Pr{ηl,1 (nl ) + ηl,2 (nl ) + .... = nl } = 1, and hence ηl (nl ) should be viewed
as r.vec. of frequencies in the random allocation of nl particles into infinitely many cells
labeled by {1, 2, ...}. The r.v.ηl,m (nl ) then is the number of particles falling into the mth cell after allocation of all nl particles. The independent r.vec.s η1 (n1 (n1 ), ..., ηs (ns ) all
together can be viewed, for example, as a random allocation of particles of s types, where
nl is the number of particles of the l - th type, as well as random allocation of particles
in sets, where now nl is the number of particles of l -th set. Our goal here is to present
a unified approach to derive asymptotic (as min(n1 , ..., ns ) → ∞ ) approximation for
distribution function of general classes of statistics of the form
R(n) =∞X

fm (η1,m , ..., ηs,m ),m=1where fm (x1 , ..., xs ) is a sequence of functions (may be a random) defined for x1 ≥0, ..., xs ≥ 0, such that the series R(n) is a.s. converges for every n = (n1 , ..., ns ). We
80

emphasize that the allocation scheme is determined by the distribution of r.v,s ξl,m (nl ).
The generalized urn model and statistics considered by Miakhmedov et al (2014) follows
if s = 1 and P r{ξ1,m (n1 ) = 0} = 1 for m > N. some N = N (n) → ∞.
The following examples are most often encountered in applications.
(i) Independent random allocation scheme. Into an infinite number of cells,labelledby {1, 2, ?.} , we throw particles of s types, where nk particles of k -th type, k = 1, ..., s.
The particles are thrown randomly one by one and independently of each other. The
probability of a particle of k-th type falling into m-th cell is pk,m ≥ 0 and pk,1 +pk,2 +... = 1. This model determines by =(ξl,m (nl )) = P oi(nl pl,m ) . For instance, the number ofoccupied cells and the number of cells containing exactly a given number of particles
of each types are examples of R(n). The special case when pl,m > 0 , m = 1, ..., N andpl,m = 0 for all m > N , l = 1, ..., s, some N = N (n1 , ..., ns ) → ∞ as min(n1 , ..., ns ) → ∞,is well studied in the literature multinomial random allocation model, alternatively
known as a sample scheme with replacement from finite population of size N. The classicalchi-square, likelihood-ratio statistic, and the empty-cells statistic are examples of RN (n). Here and below RN (n) stands for the statistics R(n) , where P r{ξl,m (nl ) = 0} = 1 forall m > N ,l = 1, ..., s .
(ii) Sample scheme without replacement from a finite population . Assume from astratified population of size ΩN = ω1 + ... + ωN , where ωm > 0 is the size of m-th stratum,s independent samples of sizes n1 , ..., ns , respectively, are drawn. Each sample is carriedout according to the sample scheme without replacement. All CΩnkN possible variants of
−1
the k th sample has the same probability equal to CΩnkN. This urn model corresponds
to the case where the r.vec.ηl (nl ) has the multivariate hyper-geometric distribution. Here=(ξl.m (nl ) = Bi(ωm , nl /ΩN ), the binomial distribution with indicated parameters .For
instance the number of untouched strata and the sample sum are important variants of
RN (n). Note that the random allocation of particles in sets also is a variant of this model.
References:
1. Kolchin V.F., Sevast’yanov, B.A., and Chistyakov, V.P. (1978). Random Allocations,
V.H. Winston and Sons, Washington, DC.
2. Kotz S. and Balakrishnan N., (1997). Advances in urn models during the past two
decades. In Advances in Combinatorial Methods and Appl. to Probab. and Statist.,pp.
203-257. Birkhauser, Boston. MA.
3. Mirakhmedov S.M. , Jammalamadaka S.R., Ibrahim B. M (2014), On Edgeworth
Expansions in Generalized Urn Models.. J Theor Probab v. 27, pp.725 - 753 .

81

On the identity of the theta functions
Mizomov Inomjon
V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences
mizomovinomjon@mail.ru

Keywords: lattice, holomorphic function, theta function, algebraic surface.
Let Λ be an integral lattice generated by 1 and η ∈ C, where Im η > 0, and let E := C/Λ
be the corresponding elliptic curve. We write Θn (Λ) for the set of holomorphic functions
f on C satisfying the quasi-periodicity conditions
f (z + 1) = f (z)f (z + η) = −e−2πinz f (z) .
The functions in Θn (Λ) are called theta functions of order n with the respect to the lattice
Λ. It is well-known (see e.g. [2]), Θn is a vector space of dimension n. Next, one can easily
check that the functionXn(n−1)
θ(z) =(−1)n e2πi(nz+ 2 η)n∈Z

forms a basis for Θ1 . For α ∈ Z/nZ, we define
2πi(αz+[α])θα (z) := e

n−1
Y


m α 
θ z+
+ η ,
n
n
m=0

[α] :=α(α − n)
α
η+ .
2n
2n

(1)

Then {θ0 , θ1 , ..., θn−1 } form a basis for Θn ( see [1,Lemma 2.5] or [3, Appendix A]).
Let n = 4. Then we can rewrite (1) as

 
 

1
1
3
θ0 (z) = θ(z)θ z +
θ z+
θ z+
,
4
2
4

 
 
 

1
3
1
1 1
1 1
3 1
θ1 (z) = θ z + η θ z + + η θ z + + η θ z + + η e2πi(z+ 8 − 8 η) ,
4
4 4
2 4
4 4

 
 
 

1
1
1
1 1
1 1
3 1
θ2 (z) = θ z + η θ z + + η θ z + + η θ z + + η e2πi(2z+ 4 − 2 η) ,
2
4 2
2 2
4 2

 
 
 

3
3
3
1 3
1 3
3 3
θ3 (z) = θ z + η θ z + + η θ z + + η θ z + + η e2πi(3z+ 8 − 8 η) .
4
4 4
2 4
4 4
Our first main result is
Theorem 1. For any τ ∈ C − 41 Λ the above functions satisfy the following identityθ02 (τ ) + θ22 (τ )
θ2 (τ ) + θ32 (τ )= 1.
θ1 (τ )θ3 (τ )
θ0 (τ )θ2 (τ )(2)

If we introduce the following denotation
a :=

i
1
1
1
, b :=, c :=, d :=.
θ0 (τ )θ3 (τ )
θ0 (τ )θ1 (τ )
θ2 (τ )θ3 (τ )
θ0 (τ )θ2 (τ )we can easily get the identity a2 b3 c − b3 c3 − a2 d4 + b2 d4 = 0.
Theorem 2. The set V := {(a, b, c, d) ∈ P3 | a2 b3 c − b3 c3 − a2 d4 + b2 d4 = 0, a 6= 0, b 6=0, c 6= 0, d 6= 0} is a normal affine algebraic surface.
82References:
1. A. Chirvasitu, R. Kanda and S. P. Smith, Feigin and Odesskiivs elliptic algebras,
Journal of Algebra 581 (2021), 173–225.
2. D. Mumford, Tata lectures on theta. I, Modern Birkhauser Classics, Birkhauser Boston,Inc., Boston, MA, 2007, With the collaboration of C. Musili, M. Nori, E. Previato and
M. Stillman, Reprint of the 1983 edition.
3. A. V. Odesskii, Elliptic algebras, Uspekhi Mat. Nauk 57 (2002), no. 6(348), 87-122.

Dynamical system of an infinite-dimensional operator
in an invariant set
Utkir Rozikov, Umrbek Olimov
V.I.Romanovskiy Institute of Mathematics, Tashkent, Uzbekistanrozikovu@mail.ru, umrbek.olimov.92@mail.ru

Keywords: Infinite dimensional operator, limit point, dynamical system.
onP∞
1
Denote l+ = x = (x1 , x2 , . . . , xn , . . . ) : xi > 0, kxk = j=1 xj < ∞ .
Following [1] we consider discrete-time, infinite-dimensional dynamical systems (IDDS)
1
asgenerated by operator F defined on l+
!2
!2
P∞
P∞
x
1
+
x
1
+
2j−1
2j
j=1
j=1
F : x02n−1 = λ2n−1 ·, x02n = λ2n ·1 + θ + kxk
1 + θ + kxk
1
where n = 1, 2, . . . , θ > 0 and λ = (λ1 , λ2 , . . . ) ∈ l+.
The main problem for such a dynamical system (see Chapter 1 of [2]) is to study
1.
trajectory t(m) = F m (t(0) ), m ≥ 1 for any t(0) ∈ l+
We have proved the following
1
1
Lemma 1. If λ = (λ1 , λ2 , . . . ) ∈ l+then F maps l+
to itself.
Define two-dimensional operator W : z = (x, y) ∈ R2+ → z 0 = (x0 , y 0 ) = W (z) ∈ R2+ by

2

2
1+y
1+x
0
0
W : x = L1 ·, y = L2 ·,1+θ+x+y
1+θ+x+ywhere θ > 0 and Li > 0 are parameters.
Lemma 2. The IDDS generated by the operator F is fully represented by the twodimensional DS generated by the operator W .
DenoteM− = {(x, y) ∈ R2+ : x < y},M0 = {(x, y) ∈ R2+ : x = y},
M+ = {(x, y) ∈ R2+ : x > y}.
Lemma 3. If L1 = L2 = L then the sets M ,  = −, 0, + are invariant with respect tooperator W , i.e., W (M ) ⊂ M .
83

Reduce the operator W defined by on the invariant set M0 , then we get
2

1+x
0
.
x = f (x) := L
1 + θ + 2x

(1)

Lemma 4. The types of fixed points are as follows
1) The unique fixed pointattracting, if θ > 17, L ∈/ (L̂1 , L̂2 ),
saddle, if θ = 17, L = 108,x1 = attarcting, if θ = 17, L 6= 108 or θ ∈ (0, 1) ∪ (1, 17)2) If θ > 17 and L = L̂1 (resp. L = L̂2 ) then the function f has two fixed points x1 < x2
and x1 is saddle and x2 is attracting (resp. x1 is attracting and x2 is saddle).
3) If θ > 17, L ∈ (L̂1 , L̂2 ) then f has three fixed points with x1 < x2 < x3 . Moreover,x1 and x3 are attracting and x2 is repelling;where√2θ2 + 76θ − 142 − (2θ − 34) θ2 − 18θ + 17
L̂1 =
,
16
√
2θ2 + 76θ − 142 + (2θ − 34) θ2 − 18θ + 17
.
L̂2 =
16
The following is main result. The following theorem describes all limit points on M0 .
Theorem 1. The following assertions hold1) If θ ∈ (0, 17], L > 0 or θ > 17, L ∈/ (L̂1 , L̂2 ) then for any x ∈ (0, +∞) the following
equality holds
lim f n (x) = x1 .
n→∞2) If θ > 17 and L = L̂1 (resp. L = L̂2 ) then(x1 , if x ∈ (0, x1 ],n
lim f (x) =n→∞x2 , if x ∈ (x1 , +∞);
(
!
x
,
if
x
∈
(0,
x
),
1
2
resp. lim f n (x) =n→∞x2 , if x ∈ [x2 , +∞);
3) If θ > 17, L ∈ (L̂1 , L̂2 ) then
x , if x ∈ (0, x2 ), 1n
x2 , if x = x2 ,lim f (x) =n→∞x3 , if x ∈ (x2 , +∞).
References:
1. Olimov U.R., Rozikov U.A., Fixed points of an infinite dimensional operator related to
Gibbs measures, Theor. Math. Phys. Vol. 214, No.2, 2023, pp. 282-295.
2. Rozikov U.A., An introduction to mathematical billiards. World Sci. Publ. Singapore.
2019, 224 p.
84

Classification of non-strongly nilpotent filiform Leibniz
algebras of dimension 12
A. M. Sattarov
UNIVERSITY OF BUSINESS AND SCIENCE
e-mail: saloberdi90@mail.ru

Keywords: filiform Leibniz algebra, characteristic nilpotent, strongly nilpotent, derivation.
Leibniz algebra was first introduced in the early 90’s of the last century by Frenchmathematician J.-L.Lodey [2], [3] as a non-associative algebra with multiplication
satisfying the identity:[x, [y, z]] = [[x, y], z] − [[x, z], y].
The class of filiform algebras are most investigated subclass of nilpotent Leibniz
algebras. In [1,5] it is shown that the class of all filiform Leibniz algebras is split into
three non-intersected families, where one of the families contains filiform Lie algebras andthe other two families come out from naturally graded non-Lie filiform Leibniz algebras.
Moreaver, an isomorphism eriterion for these two families of filiform Leibniz algebras
have been given in [5]. There are series works which devoted to the classification offiliform Leibniz algebras with given dimensional. Nowadays all filiform Leibniz algebras areclassified up to dimension ten (see [4,6,7]and others).In This work using the this criteria,
we give the classification of non-strongly nilpotrnt Leibniz algebras for the first and thesecond classes of demension 12.
Definition 1. A linear transformation P of the Leibniz algebra L is called a prederivation if for any x, y, z ∈ L,
P ([[x, y], z]) = [[P (x), y], z] + [[x, P (y)], z] + [[x, y], P (z)].
It is obvious that any derivation of L is a pre-derivation.
For the given Leibniz algebra L we consider the following lower central series
L1 = L,Lk+1 = [Lk , L1 ],k ≥ 1.

Definition 2. A Leibniz algebra L is called nilpotent if there exists s ∈ N such thatL = 0.
Definition 3. A nilpotent Leibniz algebra is called characteristically nilpotent if allits derivations are nilpotent. We say that a Leibniz algebra is strongly nilpotent if any
pre-derivation is nilpotent.
Definition 4. A Leibniz algebra L is said to be filiform if dim Li = n − I, wheren = dim L and 2 ≤ I ≤ n.
The following theorem divides all n-dimensional filiform Leibniz algebras into threefamilies.
Theorem 1. Any n-dimensional complex filiform Leibniz algebra admits a basis{e1 , e2 , . . . , en } such that the table of multiplication of the algebra has one of the following
forms:s85[e1 , e1 ] = e3 ,





[ei , e1 ] = ei+1 ,2 ≤ i ≤ n − 1,



n−1
P
F1 (α4 , . . . , αn , θ) =[e1 , e2 ] =αt et + θen ,


t=4

n

P



αt−j+2 et , 2 ≤ j ≤ n − 2.
 [ej , e2 ] =t=j+2[e1 , e1 ] = e3 ,[e , e ] = ei+1 ,3 ≤ i ≤ n − 1,


 i 1
n
P
F2 (β4 , . . . , βn , γ) =[e1 , e2 ] =βt e t ,[e2 , e2 ] = γen ,


t=4

n

P


βt−j+2 et , 3 ≤ j ≤ n − 2,
 [ej , e2 ] =t=j+2[ei , e1 ] = ei+1 ,2 ≤ i ≤ n − 1,





3 ≤ i ≤ n − 1,
 [e1 , ei ] = −ei+1 ,[e1 , e1 ] = θ1 en ,[e1 , e2 ] = −e3 + θ2 en ,[e2 , e2 ] = θ3 en ,F3 (θ1 , θ2 , θ3 ) =[ei , ej ] = −[ej , ei ] ∈ hei+j+1 , ei+j+2 , . . . , en i, 2 ≤ i < j ≤ n − 1,[ei , en+1−i ] = −[en+1−i , ei ] = α(−1)i+1 en ,2 ≤ i ≤ n − 1,
where all omitted products are equal to zero and α ∈ {0, 1} for even n and α = 0 for oddn.
We give the classification of non-strongly nilpotent complex filiform Leibniz algebras
of dimension 12.
Theorem 2. Let L be a 12-dimensional non-strongly nilpotent complex filiform Leibniz
algebra.Then L is isomorphic to one of the following algebras:
F1 (0, 1, 0, 0, 0, α9 , 0, α11 , α12 , θ), F1 (0, 1, 0, −3, 0, 0, 0, α11 , α12 , θ),
F1 (0, 1, 0, −3, 0, 12, 0, 0, α12 , θ), F1 (0, 1, 0, −3, 0, 12, 0, −55, 1, θ),
F1 (0, 1, 0, −3, 0, 12, 0, −55, 0, 1), F1 (0, 1, 0, −3, 0, 12, 0, −55, 0, 0),
F1 (0, 0, 0, 1, 0, 1, 0, α11 , α12 , θ), F1 (0, 0, 0, 1, 0, 0, 0, 1, α12 , θ), F1 (0, 0, 0, 1, 0, 0, 0, −5, 1, θ),
F1 (0, 0, 0, 1, 0, 0, 0, −5, 0, 1), F1 (0, 0, 0, 1, 0, 0, 0, −5, 0, 0), F1 (0, 0, 0, 0, 0, 1, 0, 1, α12 , θ),
F1 (0, 0, 0, 0, 0, 1, 0, 0, 1, θ), F1 (0, 0, 0, 0, 0, 1, 0, 0, 0, 1), F1 (0, 0, 0, 0, 0, 1, 0, 0, 0, 0),
F1 (0, 0, 0, 0, 0, 0, 0, 1, 1, θ), F1 (0, 0, 0, 0, 0, 0, 0, 1, 0, 1), F1 (0, 0, 0, 0, 0, 0, 0, 1, 0, 0),
F1 (0, 0, 0, 0, 0, 0, 0, 0, 1, 0), F1 (0, 0, 0, 0, 0, 0, 0, 0, 1, 1),
F1 (1, −2, 5, −14, 42, −132, 429, −1430, 4862, 0), F1 (0, 0, 0, 0, 0, 0, 0, 0, 0, 0),
F1 (1, −2, 5, −14, 42, −132, 429, 0, α12 , θ), F1 (0, 0, 1, 0, 0, −4, 0, 1, α12 , θ),F1 (1, −2, 5, −14, 42, −132, 429, −1430, 0, θ), F1 (0, 0, 1, 0, 0, −4, 0, 0, 0, θ),
F1 (0, 0, 0, 0, 0, 0, 0, 0, 0, 1), F1 (0, 0, 1, 0, 0, −4, 0, 0, 22, 0),
F1 (1, −2, 5, −14, 42, −132, 429, −1430, 4862, 4862), F1 (0, 0, 1, 0, 0, −4, 0, 0, 22, 22),F1 (0, 0, 0, 0, 1, 0, 0, 1, α12 , θ), F1 (0, 0, 0, 0, 1, 0, 0, 0, 1, θ), F1 (0, 0, 0, 0, 1, 0, 0, 0, 0, 1),
F1 (0, 0, 0, 0, 1, 0, 0, 0, 0, 0), F1 (0, 0, 0, 0, 0, 0, 1, 1, α12 , θ), F1 (0, 0, 0, 0, 0, 0, 1, 0, 1, θ),
F1 (0, 0, 0, 0, 0, 0, 1, 0, 0, 1), F1 (0, 0, 0, 0, 0, 0, 1, 0, 0, 0), F2 (0, 1, 0, 0, 0, β9 , 0, β11 , β12 , 1),
86

F2 (0, 1, 0, 0, 0, 1, 0, β11 , β12 , 0), F2 (0, 1, 0, 0, 0, 0, 0, 1, β12 , 0), F2 (0, 1, 0, 0, 0, 0, 0, 0, 1, 0),
F2 (0, 1, 0, 0, 0, 0, 0, 0, 0, 0), F2 (0, 0, 0, 1, 0, 1, 0, 0, β12 , γ), F2 (0, 0, 0, 1, 0, 1, 0, 0, 0, γ),
F 2(0, 0, 0, 1, 0, 0, 0, 0, 1, γ), F2 (0, 0, 0, 0, 0, 1, 0, β11 , 0, 1), F2 (0, 0, 0, 0, 0, 1, 0, 1, β12 , 0),
F2 (0, 0, 0, 0, 0, 1, 0, 0, 1, 0), F2 (0, 0, 0, 0, 0, 1, 0, 0, 0, 0), F2 (0, 0, 0, 0, 0, 0, 0, 1, 0, 1),
F2 (0, 0, 0, 0, 0, 0, 0, 1, 1, 0), F2 (0, 0, 0, 0, 0, 0, 0, 1, 0, 0), F2 (0, 0, 0, 0, 0, 0, 0, 0, 0, 1),
F2 (0, 0, 0, 0, 0, 0, 0, 0, 1, 0), F2 (0, 0, 0, 0, 0, 0, 0, 0, 0, 0), F2 (1, 0, 0, 0, 0, 0, 0, 1, β12 , γ),F2 (1, 0, 0, 0, 0, 0, 0, 0, 1, γ), F2 (1, 0, 0, 0, 0, 0, 0, 0, 0, 1), F2 (1, 0, 0, 0, 0, 0, 0, 0, 0, 0),
F2 (0, 0, 1, 0, 0, 0, 0, 1, β12 , γ), F2 (0, 0, 1, 0, 0, 0, 0, 0, 1, γ), F2 (0, 0, 1, 0, 0, 0, 0, 0, 0, 1),
F2 (0, 0, 1, 0, 0, 0, 0, 0, 0, 0), F2 (0, 0, 0, 0, 1, 0, 0, 1, 0, γ), F2 (0, 0, 0, 0, 1, 0, 0, 1, β12 , 0),
F2 (0, 0, 0, 0, 1, 0, 0, 0, 0, γ), F2 (0, 0, 0, 0, 1, 0, 0, 0, 1, 0), F2 (0, 0, 0, 0, 0, 0, 1, 1, 0, γ),
F2 (0, 0, 0, 0, 0, 0, 1, 1, β12 , 0), F2 (0, 0, 0, 0, 0, 0, 1, 0, 0, γ), F2 (0, 0, 0, 0, 0, 0, 1, 0, 1, 0),References:
1. Ayupov S.A., Omirov B.A., On some classes of nilpotent Leibniz algebras, Siberian
Math. J. 42 (1) (2001) pp 15-24.
2. Loday J.L., Une version non commutative des alg’e bres de Lie: les alg’e bres de Leibniz,
Ens. Math.,v.39,1993,pp 269-293.
3. Loday J.L. and Pirashvili T., Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann., 296 (1993), pp 139-158.
4. Omirov B.A., Classification of eight-dimensional complex filiform Leibniz algebras,
Uzbek. Mat Zh.(3)(2005),pp 63-71.
5. Gomez J.R., Omirov B.A., On classification of complex filiform Leibniz algebras.
Algebra Colloquium. 22(1)(2005),pp 757-774.
6. Rakhimov I.S., Sozan J.,Description of nine simensional complex filiform Leibniz
algebras arising from narurally graded non Lie filiform Leibniz algebras. Int. J. Algebra 3(17-20) (2009) pp 969-980.
7. Mohd Kasim S., Rakhimov I.S., Said Husain S.K., Isomorphism classes of 10dimensional filiform. Leibniz algebras June 2014AIP Conference Proceedings 1602(1):708
715 DOI: 10.1063/1.4882563

Extreme values of functionals of integral form
Sharakhmetov Shoturgun
Tashkent State University of Economics, Tashkent, Uzbekistan.
sh.sharakhmetov@gmail.com

Keywords: Extreme value, functional, convex function.
In theoretical and practical problems of mathematics and other fields of science there
are problems of finding extreme values of functionals . The article is devoted to finding
the exact upper bounds of the functional Ef (ξ1 + ξ2 + ... + ξn ).
87

Let F (x1 , x2 , ..., xn ) be the joint distribution of arbitrarily dependent random variables
ξi ∈ [0, 1], i = 1, n. Denote by F the class of all distributions with fixed mathematical
expectations:
F = {F (x1 , x2 , ..., xn ) : Eξ1 = m1 , ..., Eξn = mn } .
Without loss of generality, we assume that m1 ≤ m2 ≤ ... ≤ mn .
Theorem. Let f be the convex increasing function. Then
sup Ef (ξ1 + ξ2 + ... + ξn ) =
F ∈F

n
X

f (n − k)(mk+1 − mk ),k=0

where m0 = 0, mn = 1. The supremum is reached on binomial distributed randomvariables B1 , B2 , ..., Bn :P (B1 = 1, B2 = 1, ..., Bn = 1) = m1 ,P (B1 = 0, B2 = 0, ..., Bn = 0) = 1 − mn ,P (Bi1 = 0, ..., Bil = 0, Bil+1 = 1, ..., Bin = 0) = mil+1 − mil , l = 1, 2, ..., n − 1.
The probability values of the remaining sets 0, 1 equal to zero.
A similar statement is true when the function is f monotonically decreasing.

On the maximum of dependent random variables
Sharipov Olimjon1,2 , Kobilov Utkir1 .
12

National University of Uzbekistan named after Mirzo Ulugbek, Tashkent, Uzbekistan.
V.I. Romanovskiy Institute of Mathematics of the Academy of Sciences of the Republicof Uzbekistan.Tashkent. Uzbekistan.
osharipov@yahoo.com, kobilov.utkir25@gmail.com

Keywords: maximum, weakly dependent random variables, limit theorems.
The distributions of the maximum of random variables have been studied by many
authors. In the books [1]-[3], the results and methods for the cases of independent andweakly dependent random variables are given.
We consider stationary sequences of random variables {Xn , n ∈ N } which satisfy
certain conditions. We assume that there exists another sequence {ξn , n ∈ Z} of randomvariables such that
Xn = f ({ξn+i , i ∈ Z}) , n ∈ N,
(1)
where f : RZ → R is a measurable function. Denote Mn = max Xi .
1≤i≤n

Our goal is to consider (1) when {ξn , n ∈ Z} satisfies some weakly dependentconditions.
Namely, we assume that {ξn , n ∈ Z} is ψ−mixing. Coefficients of ψ−mixing are defined
as following
P (AB) − P (A) P (B)l∞: A ∈ F−∞ , B ∈ Fl+k , l ∈ N, P (A) P (B) > 0
ψ (k) = sup
P (A) P (B)88where Fba is σ−field algebra generated by random variables ξa , ... , ξb . A sequence ofrandom variables {ξn , n ∈ Z} is called satisfying the ψ−mixing condition, if ψ(k) → 0,
as k → ∞.
Our aim is to establish limit distribution of max Xi . We will assume that there exist
1≤i≤nmeasurable functions f (m) (ξ−m , ..., ξ0 , ..., ξm ) : R2m+1 → R, m = 1, 2, ... such that thefollowing take place
lim nP Xn(m) > an + bn x = u(x), m ≥ m0 for some m0 ≥ 0
(2)
n→∞

(m)

where {an , n ≥ 1}, {bn , n ≥ 1} are sequences of some constants, Xnf (m) (ξ−m , ..., ξ0 , ..., ξm ) and 0 < u(x) < ∞ on some interval of positive length,


1
(m)
lim sup nP
X1 − X 1> ε → 0 as m → ∞ for any ε > 0.
bn
n→∞
(m)

Denote Mn(m)= max Xi
1≤i≤n

=

(3)

, Mn = max Xi .
1≤i≤n

Now we can formulate our main result.
Theorem. Let {Xn , n ∈ N } be a sequence of the form (1) with a stationary ψ−mixing
sequence {ξi , i ∈ Z}. Assume that conditions (2), (3) hold.
Then for all x ∈ R
P (Mn < an + bn x) → H(x) as n → ∞where H(x) = e−u(x) and e−∞ .
References:
1. Leadbetter M., Lindgren G., Rootzen H. Extremes and related properties of random
sequences and processes. Springer-Verlag, New York Heidelberg Berlin, (1983).
2. Galambos J. The asymptotic theory of extreme order statistics. Robert E. Krieger
publishing company Malabar, Florida, (1987).
3. Embrechts P., Kluppelberg C., Mikosch T. Modeling extreme events for insurance andfinance. Springer Verlag, (1997).

Central limit theorem for ρ-mixing random variables
with values in Lp[0, 1]
Sharipov O. Sh.1,2 , Muxtorov I. G’.2
12

National university of Uzbekistan. Tashkent. Uzbekistan
V.I. Romanovskiy Institute of Mathematics of the Academy of Sciences of the Republicof Uzbekistan. Tashkent. Uzbekistan
osharipov@yahoo.com, ibrohimmuxtorov199702@gmail.com

Keywords: Central limit theorem, mixing, Lp [0, 1] space.
89

Central limit theorems in Banach spaces are well studied in the case of independent
identically distributed random elements (see [1]). Our goal is to establish a central limit
theorem for weakly dependent random variables with values in Lp [0, 1] space.
We say that a sequence {Xi (t), i ≥ 1} of centered random variables in Lp [0, 1] satisfies
central limit theorem if the following weak convergence holds:n

1 X√
Xi (t) ⇒ N (t)n i=1where N (t) is some Lp [0, 1]-valued Gaussian random variable with mean zero.
We will assume that {Xn (t), n ≥ 1} satisfies ρ-mixing condition. For the sequence ofLp [0, 1]-valued random variables {Xn (t), n ≥ 1} ρ-mixing coefficients are defined as:|E (ξ − Eξ) (η − Eη)|∞kρ(n) = sup: ξ ∈ L2 Fk+n , η ∈ L2 F1 , k ∈ N
E 1/2 (ξ − Eξ)2 E 1/2 (η − Eη)2
where =ba -σ-field generated by random variables Xa , ..., Xb and L2 Fab -a family of square
integrable Fab -measurable random variables.
We say that {Xi (t), i ≥ 1} is ρ-mixing, if ρ(n) → 0 as n → ∞.
Our main result is the following
Theorem. Let {Xi (t), i ≥ 1} be a strictly stationary sequences of random variables
with values in Lp [0, 1], p ≥ 2 and the following conditions hold:EX1 (t) = 0,
E |X1 (t)|r < ∞,∞X2

ρ q (2n ) < ∞, ρ(1) < 1,n=1

for some r > p ≥ q ≥ 2
E |X1 (t + h) − X1 (t)|q → 0 as h → 0.
Then {Xi (t), i ≥ 1} satisfies central limit theorem.
References:
1. Ledoux M., Talagrand M. Probability in Banach Spaces: Isoperimetry and Processes,
Springer-Verlag, 1991, 480.

90

Central limit theorem for a 1-order autoregressive
process with random coefficients
T. M. Zuparov
Tashkent National University of Uzbek Language and Literature, Tashkent, Uzbekistan
zuparovtalat@navoiy-uni.uz

Keywords: Autoregressive processes, the strong law of large numbers, the central limit
theorem, Lindeberg condition.
Let be a {ρn , n ∈ Z}–sequence of independent, identically distributed random variables.
A sequence {Xn , n ∈ Z} that satisfies equation
Xn − µ = ρn (Xn−1 − µ) + εn ,

n∈Z

(1)

is called an autoregressive process with random coefficients, where µ = EXn and{εn , n ∈ Z}–white noise: a sequence of independent identically distributed randomvariables with mathematical expectations equal to zero and unit variances. If the sequence{ρn } satisfies condition supn |ρn | < 1 and does not depend on sequence {ξn }, then there
exists a unique strictly stationary solution of equation (??) such that EXn = µ. Thissolution has the form∞X
Ank εn−k , n ∈ Z(2)
Xn = µ +
k=0

where
Ank =

n−k+1
Y

ρj ,

k ≥ 1,

An0 = 1j=n

and the series converges a.s.
The paper [1] considers a class of autoregressive processes of the 1–order with randomcoefficients taking values in the Hilbert space. Limit theorems for this class are obtained:
the strong law of large numbers, the central limit theorem (c.l.t.), the compact law of theiterated logarithm. exponential inequalities, as well as rates of convergence.
In this paper, we consider an autoregressive process of the 1–order with randomcoefficients, which is not included in the class from [1]:Xn = νXn−1 + ξn ,

n∈Z

(3)

where 0 < ν < 1–random variable independent of the innovation sequence {ξk }. Thereexists (see, for example, [2]) a unique strictly stationary solution of equation (3) such that
EXn = 0. This solution has the form
Xn =

∞
X

ν k εn−k ,

n∈Z

k=0

Let Xk=∞P

Aj εk−j ,n∈Z–linear process with random coefficients

j=0{Ak , k = 0, 1, . . . } generated by a sequence {ξl , l ∈ Z} of independent random variablesnP
and
Xk . We use the following decomposition of the linear process Xn .
k=1

91

∞
P

Lemma 1. If the series

Ak converges absolutely, then decomposition

k=0
∞
X

Xn =

!
Aj

ξn +

j=0

takes place, where γj =

∞
P

∞
X

γj ξn−j −

j=1

∞
X

γj ξn−j+1

(4)

j=1

Ak .

k=j

Equality (4) can be proved directly by comparing the coefficients in front of the randomvariables ξl ; l = 0, ±1, ±2, . . . in the expression of the random variable Xn . In particular,from expansion (4) we get the following expansion for the sum of the first n terms of thelinear process:
! n
∞
∞
X
X
X
Sn =
Aj
ξt +
γj (ξn−j − ξn−j+1 )
(5)
j=0

t=1

j=1

Using expansion (5), in this paper we obtain a proof of the following theorem.
∞Pν j ξk−j a linear process with random coefficients satisfies
Theorem 1. Let Xk =j=0conditions
1) Ak = ν k , 0 < ν < 1. Random variable ν does not depend on {ξk , k ∈ Z};2) {ξk } a sequence of independent random variables, EXk = 0, EXk2 = σk2 < ∞, k ≥ 0;
n
P
σ2
σk2 .
3) max k2 → 0 as n → ∞, where Bn2 =0≤k≤n Bn
k=1
Then the sequence (1−ν)Sn Bn satisfies the c.l.t. if and only if the Lindeberg conditionis satisfied:n

1 X 2
Eξ I {|ξk ≥ ε|} −→ 0 as n → ∞ for any positive ε.
Bn2 k=1 k

(L)

∞
P

ν j ξk−j a linear process with constantj=0coefficients satisfies conditions 2) and 3). Then the sequence (1 − ν)Sn Bn satisfies thec.l.t. if and only if the Lindeberg condition is satisfied:
Remark 1. Let ν ∈ (0, 1) and Xk =

n

1 X 2
Eξ I {|ξk ≥ ε|} −→ 0 as n → ∞ for any positive ε.
Bn2 k=1 k(L)In the following theorem, we obtain an estimate for the rate of convergence in the c.l.t.
for more general linear processes with random coefficients.
P∞
Theorem 2. Let Xk =j=0 Aj ξk−j a linear process with random coefficients{Ak , k = 0,
1,
.
.
.
},
generatedbya sequence of {ξl , l ∈ Z} independent random variables
Pn
and Sn = k=1 Xk . If conditions are satisfies
s
∞
∞
P
P
1) EA2j = a2j < ∞; A =Aj , |A| > 1; E
kAk = bs < ∞;
j=0

k=1

2) Eξk = 0, E |ξk |s = βsk < ∞;3) {Aj } does not depend on sequence {ξk }, then the following estimate holds:92
4n = sup P
x

1
1

s+1
2s−1
1 bs
(βs0 + βsn ) s+1
Sn
s+1
s+1
.
π
− Φ(x) ≤ C(s)L(s) + 2
s/(s+1)
ABn ≤ x
Bn

References:
1. S .Bonkhar, T .Mourid. Limit theorems for hilbertian avtoregressive processes withrandom coefficients. Annales de IISUP, 2018, 62(3), pp.59-74.
2. D .Bosq. Linear Processes in Function Spaces. Theory and Applications, Springer-Verlag
New York, Inc, 2000.

93