Clustering in a stochastic model of one-dimensional gas
The Annals of Applied Probability
2008, Vol. 18, No. 3, 1026–1058
DOI: 10.1214/07-AAP481
c© Institute of Mathematical Statistics, 2008
CLUSTERING IN A STOCHASTIC MODEL OF
ONE-DIMENSIONAL GAS
By Vladislav V. Vysotsky1
St. Petersburg State University
We give a quantitative analysis of clustering in a stochastic model
of one-dimensional gas. At time zero, the gas consists of n identical
particles that are randomly distributed on the real line and have zero
initial speeds. Particles begin to move under the forces of mutual
attraction. When particles collide, they stick together forming a new
particle, called cluster, whose mass and speed are defined by the laws
of conservation.
We are interested in the asymptotic behavior of Kn(t) as n→∞,
where Kn(t) denotes the number of clusters at time t in the system
with n initial particles. Our main result is a functional limit theorem
for Kn(t). Its proof is based on the discovered localization property
of the aggregation process, which states that the behavior of each
particle is essentially defined by the motion of neighbor particles.
1. Introduction.
1.1. Description of the model. We give a quantitative analysis of clus-
tering in a stochastic model of one-dimensional gas. At time zero, the gas
consists of n point particles, each one of mass 1
. These particles are ran-
domly distributed on the real line and have zero initial speeds. Particles
begin to move under the forces of mutual attraction. When two or more
particles collide, they stick together forming a new particle, called cluster,
whose mass and speed are defined by the laws of mass and momentum
conservation. Between collisions, particles move according to the laws of
Newtonian mechanics.
We suppose that the force of mutual attraction does not depend on dis-
tance and equals the product of masses. This assumption is natural for
Received March 2007; revised September 2007.
1Supported in part by the Grants NSh-4222.2006.1 and DFG-RFBR 436 RUS
113/773/0-1(R).
AMS 2000 subject classifications. Primary 60K35, 82C22; secondary 60F17, 70F99.
Key words and phrases. Sticky particles, particle systems, gravitating particles, number
of clusters, aggregation, adhesion.
This is an electronic reprint of the original article published by the
Institute of Mathematical Statistics in The Annals of Applied Probability,
2008, Vol. 18, No. 3, 1026–1058. This reprint differs from the original in
pagination and typographic detail.
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2 V. V. VYSOTSKY
one-dimensional models because, by the Gauss law applied to flux of the
gravitational field, gravitation is proportional to the distance to the power
one minus dimension of the space. At any moment, the acceleration of a
particle is thus equal to difference of masses located to the right and to the
left of the particle.
Random initial positions of particles are usually described (see [8, 16,
25]) by the following natural models: in the uniform model, n particles are
independently and uniformly spread on [0,1]; in the Poisson model, particles
are located at points 1
S2, . . . ,
Sn, where Si is a standard exponential
random walk. In other words, particles are located at points of first n jumps
of a Poisson process with intensity n.
These two models are the most natural and interesting; let us call them
the main models of initial positions. However, we will see that behavior
of the Poisson model is essentially defined by independence of initial dis-
tances between particles rather than by the particular type of the distances’
distribution. Therefore, it is of a great mathematical interest to general-
ize the Poisson model by introducing the i.d. model, where “i.d.” stands
for “independent distances,” as follows. Particles are initially located at
S2, . . . ,
Sn, where Si is a positive random walk whose nonnegative
i.i.d. increments Xi satisfy the normalization condition EXi = 1. Note that
if we proceed to the limit as n→∞, we consider a system of total mass one,
which consists of, roughly speaking, infinitesimal particles homogeneously
spread on [0,1]; this is true for all the mentioned models of initial positions.
The mathematical interest in sticky particles systems arises mainly from
relations between these systems and some nonlinear partial differential equa-
tions originating from fluid mechanics, for example, the Burgers equation.
These equations admit interpretation in terms of sticky particles; see Gur-
batov et al. [10], Brenier and Grenier [4] or E, Rykov and Sinai [6]. Sticky
particles models are also used for numerical solving of other partial differen-
tial equations; see Chertock et al. [5] for explanations and further references.
As time goes, particles aggregate in clusters. Clusters become larger and
larger while the number of clusters decreases until they merge into a single
cluster containing all initial particles. This process of mass aggregation is
strongly connected with additive coalescence; see Bertoin [2] and Giraud [9]
for the most recent results and references.
The aggregation process resembles formation of a star from dispersed
space dust and sticky particles models indeed have relations to astrophysics.
It is appropriate to clarify these relations since they are not so direct and
cause a lot of misunderstanding.
It is known that the distribution of galaxies in the universe is very inho-
mogeneous and the regions of high density form a peculiar cellular structure.
The first attempt to understand the formation of such structures was made
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 3
in 1970 by Zeldovich. Most of the mass in the universe is believed to ex-
ist in the form of particles that practically do not collide with each other
and interact only gravitationally, for example, neutrinos. In his model, Zel-
dovich considered an initially homogeneous collisionless medium of particles
moving by pure inertia; the gravitational interaction was taken away by an
appropriate time change. He showed that singularities, that is, the thin re-
gions of very high density of particles, so called “pancakes,” appear even if
initial speeds of particles form a smooth velocity field.
Zeldovich’s approximate model, however, does not explain formation of
the cellular structure of matter. His approximation does not take into ac-
count that particles hitting a “pancake” are hampered by its strong gravita-
tional field and start oscillating inside the “pancake” instead of flying away.
Although this gravitational adhesion of collisionless particles is not precisely
the same as the real sticking, the model of sticky particles serves as a reason-
able approximation. The effect of gravitational adhesion was then analyzed
by the use of the Burgers equation; Gurbatov, Saichev and Shandarin pro-
posed it in 1984 to extend Zeldovich’s approximation, which is invalid after
formation of “pancakes.”
The model of sticky particles is directly mentioned in Gurbatov et al. [11];
a comprehensive survey of the formation of the Universe’s large-scale struc-
ture could be found in Shandarin and Zeldovich [23].
1.2. Statement of the problem and the results. In general, the problem
is to describe the process of mass aggregation. How fast is it? How large
the clusters are? Where do clusters appear most intensively, and so forth?
Numerous papers on the model (e.g., [8, 14, 16, 20, 25]) are dedicated to
probabilistic description of various properties of the aggregation process as
the number of initial particles n tends to infinity. Thus, the behavior of a
typical system consisting of a large number of particles is studied.
In this paper, we are interested in the asymptotic behavior of Kn(t), which
denotes the number of clusters at time t in the system with n initial particles.
This variable is a decreasing random step function satisfying Kn(0) = n
and Kn(t) = 1 for t ≥ T lastn , where T lastn denotes the moment of the last
collision. While calculating Kn(t), we also count initial particles that have
not experienced any collisions; in other words, Kn(t) is the total number of
particles existing at time t.
It is very important to know the behavior of Kn(t). This gives us a deep
understanding of the aggregation process since the average size of a cluster
at time t is n
Kn(t)
At first we give a short deterministic example. Suppose that particles are
located at points 1
, . . . , n
, that is, Si = i. By simple calculations, we find
that there would not be any collisions before t= 1. At the moment t= 1, all
4 V. V. VYSOTSKY
particles simultaneously stick together, hence Kn(t) = n for 0 ≤ t < 1 and
Kn(t) = 1 for t≥ 1.
However, when the initial positions are random, the aggregation process
behaves entirely differently. In [25], the author proved the following state-
ment.
Fact 1. There exists a deterministic function a(t) such that both in the
Poisson and the uniform models of initial positions, for any t≥ 0, we have
Kn(t)
P−→ a(t), n→∞.(1)
The function a(t) is continuous, a(0) = 1, and a(t) = 0 for t≥ 1. We conjec-
ture, on the basis of numerical simulations, that a(t) = 1− t2 for 0≤ t≤ 1.
The relation a(t) = 0 for t > 1 is not of a surprise because we know from
Giraud [8] that both in the Poisson and the uniform models, T lastn
P−→ 1 (the
limit constant is so “fine” due to the proper scaling of the model). Therefore,
we say that the moment t= 1 is critical ; note that this moment coincides
with the moment of the total collision in the deterministic model.
The aim of this paper is to strengthen the result of [25]. We first gen-
eralize Fact 1 and prove it for the i.d. model. We will see [relations (19)
and (27) below] that a(t) is equal to the probability of a certain event that
is expressed in terms of Xi. Also, we will prove that a(t) depends on the
common distribution of Xi as follows: a(t) = 1 on [0,
µ), where
µ := sup{y :P{Xi < y}= 0};
a(t) ∈ (0,1) on (√µ,1); and a(t) = 0 on (1,∞).
Furthermore, the recent results of the author [26] allow us to prove the
conjecture from Fact 1 that aPoiss(t) = aUnif(t) = 1− t2 for 0≤ t≤ 1. There
is an amazing contrast between the simplicity of this formula and the hard
calculations one needs to obtain it. It is remarkable that now we know the
limit function a(t) for the main models of initial positions.
Our main goal is to improve (1) by finding the next term in the asymp-
totics of Kn(t). The result is the following statement, where the standard
symbol
D−→ denotes weak convergence and D denotes the Skorohod space.
Theorem 1. In the i.d. model with continuous Xi satisfying EX
for some γ > 4, there exists a centered Gaussian process K(·) on [0,1) such
Kn(·)− na(·)√
D−→K(·) in D[0,1− ε] for all ε ∈ (0,1)(2)
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 5
as n→ ∞. The process K(·) depends on the distribution of Xi. This pro-
cess satisfies K(0) = 0 and has a.s. continuous trajectories. The covariance
function R(s, t) of K(·) is continuous on [0,1)2, R(s, t)> 0 on (√µ,1)2, and
R(s, t) = 0 on [0,1)2 \ (√µ,1)2.
In the uniform model, (2) holds for some centered Gaussian process KUnif(·)
on [0,1). This process satisfies KUnif(0) = 0 and has a.s. continuous trajecto-
ries. The covariance function RUnif(s, t) of KUnif(·) is continuous on [0,1)2,
and RUnif(s, t) =RPoiss(s, t)− s2t2.
Thus, the Poisson and the uniform models lead to different limit processes
KPoiss(·) and KUnif(·), although aPoiss(·) = aUnif(·).
As an immediate corollary of Theorem 1 (see Billingsley [3], Section 15),
we get
Kn(t)− na(t)√
D−→N (0, σ2(t)), n→∞(3)
for any t < 1, where σ2(t) := R(t, t). It is possible to show that in the i.d.
model, (3) holds for all t 6= 1 under the less restrictive condition EX2i <∞,
with σ2(t) = 0 for t > 1; continuity of Xi is not required.
We also study convergence of the left-hand side of (3) at the critical
moment t= 1. Apparently, the limit is not Gaussian, but this complicated
problem is related to a curious, but hardly provable conjecture on integrated
random walks. In view of this non-Gaussianity, it seems impossible to prove
any extended version of Theorem 1 that describes the weak convergence of
trajectories on the whole interval [0,1]; we refer to Section 7 for further
discussion.
We finish this subsection with a note on scaling. In our model, the masses
of particles are equal to 1
and the distances between them are of the order
. Let us rescale the i.d. model by multiplying all masses and distances
by n: the system of particles of mass one each, initially located at points
S1 − S[n/2], S2 − S[n/2], . . . , Sn − S[n/2], is called the expanding model. The
particles are shifted by S[n/2] because we want the system to expand “filling”
the whole line as n→∞ rather than only the positive half-line.
All results of our paper hold true for the expanding model. This is not
unexpected because the shift does not produce any changes and the rescaling
of masses is equivalent to the time contraction by n times while the rescaling
of distances is equivalent to the time expansion by n times. We refer the
reader to Section 2 below or to Lifshits and Shi [16] for rigorous arguments.
1.3. Organization of the paper. In Section 2 we describe a general method
which is used to study systems of sticky particles. This method is applied for
studying the i.d. model in Section 3, where we investigate some properties of
6 V. V. VYSOTSKY
the aggregation process. We will show that the aggregation process is highly
local, that is, the behavior of a particle is essentially defined by the motion
of neighbor particles. This localization property suggests that we could use
limit theorems for weakly dependent variables to prove both Fact 1 and
Theorem 1 for the i.d. model; this will be done in Section 4. Then we will
prove Theorem 1 for the uniform model in Section 5. In Section 6 we study
the number of clusters at the critical moment t = 1. Some open questions
are discussed in Section 7.
2. Method of barycenters. In this section we briefly describe the method
of barycenters, which is the main tool used to study systems of sticky par-
ticles; it is also applicable to more general models where particles could
have nonzero initial speeds and different masses. The method of barycenters
was independently introduced by E, Rykov and Sinai [6] and Martin and
Piasecki [20].
Let us start with several definitions. We always numerate particles from
left to right and identify particles with their numbers. A block of particles is
a nonempty set J ⊂ [1, n] consisting of consecutive numbers. For example,
the block (i, i+k] consists of particles i+1, . . . , i+k. Note that there are not
any relations between blocks and clusters: for example, a block’s particles
could be contained in different clusters and these clusters could even contain
particles that do not belong to the block.
It is convenient to assume that initial particles do not vanish at collisions
but continue to exist in created clusters. Then the coordinate xi,n(t) of a
particle i could be defined as the coordinate of a cluster that contains the
particle at time t. The second subscript n always indicates the number of
initial particles; we will omit this subscript as often as possible.
By xJ(t) := |J |−1
i∈J xi(t) denote the position of the barycenter of a
block J at time t. Further, define
x∗J(t) := xJ(0) +
where M
J := n
−1(n −maxj∈J) and M (L)J := n−1(minj∈J −1) are the to-
tal masses of particles located to the right and to the left of the block J ,
respectively.
A block is free from the right up to time t if, up to this time, the block’s
particles did not collide with particles initially located to the right of the
block. We similarly define blocks that are free from the left and say that a
block is free up to time t if it is both free from the right and from the left.
The next statement plays the key role in the analysis of sticky particles
systems. The barycenter of a free block moves as an imaginary particle con-
sisting of all particles of the block put together at the initial barycenter. In a
more precise and general way, we state the following.
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 7
Proposition 1. If a block J is free from the right (resp. left) up to time
t, then xJ(s)≥ x∗J(s) for s ∈ [0, t] [resp. xJ(s)≤ x∗J(s)]. If a block J is free
up to time t, then xJ(s) = x
J(s) for s ∈ [0, t].
This statement could be found, for example, in Lifshits and Shi [16],
Proposition 4.1. The easy proof is based on the property of conservation of
momentum.
The moment when a particle j sticks with its right-hand side neighbor
j + 1 is called the merging time Tj,n of the particle j. In other words, Tj,n
is the first moment when particles j and j + 1 are contained in a common
cluster; here j ∈ [1, n− 1]. Proposition 4.3 from Lifshits and Shi [16], which
is stated below, gives us a way to calculate Tj,n.
Proposition 2. For every j ∈ [1, n− 1], we have
Tj,n = min
j<k≤n
0≤l<j
{s≥ 0 :x∗(j,k](s) = x∗(l,j](s)}.(4)
Thus, Tj,n is expressed by means of barycenters. Note that since
x∗(j,k](s)− x∗(l,j](s) = x(j,k](0)− x(l,j](0)−
s2,(5)
each of the equations x∗
(j,k]
(s) = x∗
(l,j]
(s) has a unique nonnegative solution.
We also mention that at the moment Tj,n appears a cluster that consists of
the particles l+1, . . . , k, where k and l are minimizers of the right-hand side
of (4).
We will prove Proposition 2 since the proof is simple and perfectly illus-
trates the sense of the method of barycenters.
Proof of Proposition 2. For any u < Tj,n, the particles j and j + 1
are contained in different clusters. Therefore, for every l < j, the block [l, j]
is free from the right up to time u, and for every k > j, the block [j + 1, k]
is free from the left. By Proposition 1,
x∗(l,j](u)≤ x(l,j](u)≤ xj(u)<xj+1(u)
≤ x(j,k](u)≤ x∗(j,k](u),
and since, by (5), the function x∗
(j,k]
(s)− x∗
(l,j]
(s) is decreasing for s≥ 0, we
conclude that
u <{s≥ 0 :x∗(j,k](s) = x∗(l,j](s)}.
Taking minimum over k, l and taking supremum over u, we get Tj,n ≤
min{· · ·}.
8 V. V. VYSOTSKY
Let us prove the last inequality in the other direction. By the definition of
Tj,n, there exist an l < j and a k > j such that the blocks (l, j] and (j, k] are
free up to time Tj,n (clusters containing particles from these blocks collide
exactly at time Tj,n). In view of Proposition 1,
x∗(l,j](Tj,n) = x(l,j](Tj,n) = x(j,k](Tj,n) = x
(j,k](Tj,n);
hence Tj,n = {s≥ 0 :x∗(j,k](s) = x
(l,j]
(s)} and Tj,n ≥min{· · ·}. �
3. Study of the i.d. model. The localization property. At first, note that
Kn(t) = 1+
1{t<Ti,n}(6)
because the total number of clusters decreases by one at each moment Ti,n.
This representation plays the key role in the investigation of Kn(t). Clearly,
we need to study properties of the r.v.’s Ti,n to prove limit theorems for
Kn(t); such study will be done in this section.
3.1. The initial study. Let us simplify the representation for Tj,n from
Proposition 2. In this section we consider the i.d. model of initial positions,
where xj,n(0) =
Sj . Recall that Sj is a random walk with i.i.d. increments
{Xj}j∈Z (we will need the variables {Xj}j≤0 later).
Rewrite the initial distance between barycenters as
x(j,k](0)− x(l,j](0)
i=j+1
j − l
i=l+1
i=j+1
(Si − Sj+1) +
j − l
i=l+1
(Sj − Si) + (Sj+1 − Sj)
k−j−1
(Sj+i+1 − Sj+1) +
j − l
j−l−1
(Sj − Sj−i) +Xj+1
let us agree that
:= 0. Further, by
x(j,k](0)− x(l,j](0)
k−j−1
j+i+1
m=j+2
j − l
j−l−1
m=j−i+1
Xm +Xj+1
k−j−1
(k− j − i)Xj+i+1
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 9
j − l
j−l−1
(j − l− i)Xj−i+1 +Xj+1
and (5), we have
x∗(j,k](s)− x∗(l,j](s) =
Fk−j,j,j−l(s),
where
Fp,j,q(s) :=
(p− i)Xj+i+1
(q − i)Xj−i+1 +Xj+1 −
(for p, q ≥ 1 and j ∈ Z). Now, by Proposition 2, we get
Tj,n = min
j<k≤n
0≤l<j
{s≥ 0 :Fk−j,j,j−l(s) = 0}
= min
1≤k≤n−j
1≤l≤j
{s≥ 0 :Fk,j,l(s) = 0}.(8)
Note that Fp,j,q(0) ≥ 0 for all p, j, q and Fp,j,q(s) is decreasing for s ≥ 0.
This function could be also written in the more convenient form:
Fp,j,q(s) =
(p− i)(Xj+i+1 − s2)
(q − i)(Xj−i+1 − s2) + (Xj+1 − s2).
3.2. Localization property of the aggregation process. We see that Tj,n
is a function of X2, . . . ,Xn; in other words, it is necessary to know the
distances between all n particles to find Tj,n. The aggregation process is
actually highly local, that is, the value of Tj,n is essentially defined by the
initial distances between neighbor particles {i} of j for which |j − i| is small
enough.
To make this statement rigorous, we need to introduce the following no-
tation. Let us put
j := min
1≤k,l≤M
{s≥ 0 :Fk,j,l(s) = 0}, j ∈ Z,M ∈N,
which is expressed in terms of the variables {Xi}|j−i|≤M only. Also, define
Tj := inf
k,l≥1
{s≥ 0 :Fk,j,l(s) = 0}, j ∈ Z,
10 V. V. VYSOTSKY
which is, in some sense, the merging time in an appropriate infinite system
of particles. The reader could construct such system by considering the limit
of the expanding model, see Section 1.
It is clear that
Tj ≤ Tj,n ≤ T (j∧n−j)j , j, n ∈N, j ≤ n,(10)
where by ∧ and ∨ we denote minimum and maximum, respectively, and
Tj ≤ T (M)j , j ∈ Z,M ∈N.(11)
Let us estimate the rate of the convergence of P{Tj 6= T (M)j } to zero as the
“radius of the neighborhood” M tends to infinity. We thus could “measure”
the above-mentioned locality of the aggregation process. In fact, by (10), we
have P{Tj,n 6= T (M)j } ≤ P{Tj 6= T
j } for any n ∈N, j ≤ n, andM ≤ j∧n−j.
Lemma 1. Suppose EX
i <∞ for some γ ≥ 1. Then there exists a non-
decreasing function ρ(t) such that
max(P{1{t≤Tj} 6= 1{t≤T (M)
}},P{Tj 6= T
j , T
j ≤ t})≤ ρ(t)M
1−γ(12)
for any t ∈ (0,1), j ∈ Z, and M ∈N. Moreover, for any t < 1, the left-hand
side of (12) is o(M1−γ).
Proof. Let us estimate the first probability in the left-hand side of
(12). By properties of Fk,j,l(·) and definitions of T (M)j and of Tj ,
P{1{t≤Tj} 6= 1{t≤T (M)
}}= P{Tj < t≤ T
k,l≥1
Fk,j,l(t)< 0, min
1≤k,l≤M
Fk,j,l(t)≥ 0
By (9), this expression does not depend on j, and putting j :=−1,
P{1{t≤Tj} 6= 1{t≤T (M)
(k− i)(Xi − t2)
+ inf
(l− i)(X−i − t2) + (X0 − t2)< 0,
1≤k≤M
(k− i)(Xi − t2)
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 11
+ min
1≤l≤M
(l− i)(X−i − t2) + (X0 − t2)≥ 0
We then compare the inequalities in the braces and obtain
P{1{t≤Tj} 6= 1{t≤T (M)
(k− i)(Xi − t2)< min
1≤k≤M
(k− i)(Xi − t2)
(Si − it2)< min
1≤k≤M
(Si − it2)
(Si − it2)< min
k∈{1,M}
(Si − it2)
Now rewrite the event in the last line as
∃k >M : 1
(Si − it2)<min
(Si − it2)
∃k >M : 1
(Si − it2)
(Si − it2)<min
(Si − it2)
Analyzing both cases 0≤ 1
i=1 (Si − it2) and 0> 1M
i=1 (Si − it2), we
conclude that the considered event implies
∃k >M : 1
(Si − it2)< 0
∃k >M :
(Si − it2)< 0
Clearly, the latter implies
{∃i≥M :Si − it2 < 0}=
hence, combining all the estimates together, we get
P{1{t≤Tj} 6= 1{t≤T (M)
}} ≤ 2P
.(13)
Note that we obtained (13) without any assumptions on the moments of Xi.
We now estimate the right-hand side of (13); recall that EXi = 1. Then
the first part of (12) immediately follows from the classical result of Baum
and Katz [1] (see their Theorem 3 and Lemma):
12 V. V. VYSOTSKY
Fact 2. If EXi = a and E|Xi|γ <∞ for some γ ≥ 1, then
= o(k1−γ), k→∞
for any ε > 0. In addition, the series
k=1P{supi≥k |Sii − a|> ε} converges
for all ε > 0 if γ = 2.
The estimation of the second probability in the left-hand side of (12) is
completely analogous, since
{Tj 6= T (M)j , T
j ≤ t}
= {Tj < T (M)j ≤ t}
1≤k,l
Fk,j,l(T
j )< 0, min
1≤k,l≤M
Fk,j,l(T
j ) = 0, T
j ≤ t
We put j :=−1, repeat the estimates, and get
P{Tj 6= T (M)j , T
j ≤ t} ≤ 2P{∃i≥M :Si − i[T
< 0, T
−1 ≤ t}
instead of (13). The right-hand side does not exceed 2P{∃i≥M : Si − it2 <
0}, hence
P{Tj 6= T (M)j , T
j ≤ t} ≤ 2P
.(14)
3.3. The distribution function of T0 in the Poisson model. It is amazing
that in the Poisson model, the distribution function of T0 could be found
explicitly. This is important because by (27) below, the limit function a(t)
equals P{T0 > t} for the i.d. model. Also, in the proof of Theorem 1 for the
uniform model, we will need aPoiss(t) = P{TPoiss0 ≥ t} to be twice differen-
tiable and have a continuous second derivative.
Lemma 2. In the Poisson model, for 0≤ t≤ 1, we have
P{T0 ≥ t}= 1− t2.(15)
In addition, for t≥ 0, n≥ 2, and 1≤ j ≤ n− 1, we have
P{Tj,n ≥ t}= et
1≤k≤j
(Si − it2)≥ 0
1≤k≤n−j
(Si − it2)≥ 0
where Si is a standard exponential random walk.
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 13
Proof. We start with (16). By (8), (9) and properties of Fk,j,l(·),
P{Tj,n ≥ t}= P
1≤k≤n−j
1≤l≤j
Fk,j,l(t)≥ 0
1≤k≤n−j
(k− i)(Xj+i+1 − t2)(17)
+ min
1≤l≤j
(l− i)(Xj−i+1 − t2) +Xj+1 − t2 ≥ 0
In the right-hand side of the last equality, by Y denote the first minimum
and by Ỹ denote the second one.
Suppose X is a standard exponential r.v., Z is a nonnegative r.v., and
that X and Z are independent; then
P{Z ≤X}=
P{Z ≤ x}e−x dx
E1{Z≤x}e
−x dx= E
1{Z≤x}e
−x dx= Ee−Z .
Hence in view of independence of Y , Ỹ , Xj+1 we get
P{Y + Ỹ +Xj+1 − t2 ≥ 0}= EeY+Ỹ−t
EeY−t
EeỸ−t
and therefore,
P{Tj,n ≥ t}= et
P{Y +Xj+1 − t2 ≥ 0} · P{Ỹ +Xj+1 − t2 ≥ 0}.
Now, by
P{Ỹ +Xj+1 − t2 ≥ 0}
1≤l≤j
(l− i)(Xj−i+1 − t2) +Xj+1 − t2 ≥ 0
1≤l≤j
(l− i)(Xi+1 − t2) + l(X1 − t2)
1≤l≤j
(l− i+1)(Xi − t2)≥ 0
we conclude the proof of (16). Indeed, the expression in the last line equals
the first probability in the right-hand side of (16).
14 V. V. VYSOTSKY
Now let us prove (15). From the definition of T0 and T
0 we see that
1{t≤T (k)
} → 1{t≤T0} a.s. as k→∞; then by (16),
P{T0 ≥ t}= et
(Si − it2)≥ 0
Then we need to check that
(Si − it)≥ 0
1− te−t/2
for 0 ≤ t ≤ 1. The complicated calculations of this probability take more
then ten pages. Therefore, they were separated into independent paper [26].
Although these calculations seem to be technical, they are based on quite
original ideas. �
3.4. Some properties of the variables Ti. In this subsection we prove
several important properties of the r.v.’s Ti.
1. The sequence Ti is stationary.
Proof. This statement immediately follows from the definition of Ti and
stationarity of Xi, which are i.i.d.
2. The common distribution function of Ti is defined by
P{Ti ≥ t}= P
(k− i)(Xi − t2)
+ inf
(l− i)(X−i − t2) + (X0 − t2)≥ 0
Proof. This formula follows from (9).
3. We have P{√µ ≤ Ti ≤ 1} = 1 while sup{y :P{Ti < y} = 0} =
µ and
inf{y :P{Ti < y} = 1} = 1; recall that µ = sup{y :P{Xi < y} = 0}. In addi-
tion, if 0<DXi <∞, then P{Ti = 1}= 0.
Proof. First, P{√µ ≤ Ti} = 1 is trivial, because both infima in (19) are
nonpositive.
Second, fix a t≥ 1 and consider P{Ti ≥ t}. Taking into account that infima
in (19) are nonpositive, we obtain
P{Ti ≥ t} ≤ P
(k− i)(Xi − t2) + (X0 − t2)≥ 0
Then by the same arguments as in (18),
P{Ti ≥ t} ≤ P
(k − i+1)(Xi − t2)≥ 0
(Si − it2)≥ 0
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 15
By the strong law of large numbers, this probability is zero for all t > 1.
If t= 1 and 0<DXi <∞, then
(Si − i)≥ 0
= lim
1≤k≤n
(Si − i)≥ 0
= lim
1≤k≤n
Si − i√
and from the invariance principle, we get
P{Ti ≥ 1} ≤ P
0≤s≤1
W (u)du≥ 0
It follows from the asymptotics of unilateral small deviation probabilities of
an integrated Wiener process, see (43) and (44) below, that the last expres-
sion equals zero.
Third, sup{y :P{Ti < y}= 0}=
µ and inf{y :P{Ti < y}= 1} = 1 follow
if we prove that for any t < EXi = 1, the common distribution of the i.i.d.
infima in (19) has an atom at zero. But we have
(k− i)(Xi − t2) = 0
(Si − it2) = 0
and it could be shown via the strong law of large numbers that the last
probability is strictly positive for all t < 1.
4. Suppose Xi is continuous. Then T
j and Tj,n are continuous for any
j, k,n and the common distribution of Tj could have an atom only at 1. In
addition, if EX2i <∞, then Tj are continuous.
Proof. By (7) and (8),
Tj,n = min
1≤k≤n−j
1≤l≤j
H(k, j, l),(20)
where
H(p, j, q) :=
(p− i)Xj+i+1 +
(q − i)Xj−i+1 +Xj+1
Hence Tj,n is continuous as a minimum of a finite number of continuous
r.v.’s. The T
j are also continuous because T
= Tk,2k.
16 V. V. VYSOTSKY
Now we prove the continuity of Tj . By Property 3, it only remains to
verify that P{Tj ≥ t} is continuous on [0,1). But P{T (k)j ≥ t} − P{Tj ≥ t}=
P{1{t≤Tj} 6= 1{t≤T (k)
}}, and in view of (13),
0≤t≤s
|P{T (k)j ≥ t}−P{Tj ≥ t}| ≤ sup
0≤t≤s
for every s < 1 = EXi. The last expression tends to zero by the strong law
of large numbers; then P{Tj ≥ t} is continuous on [0, s] as a uniform limit
of continuous functions P{T (k)j ≥ t}. Since s < 1 is arbitrary, P{Tj ≥ t} is
continuous on [0,1).
5. The cov(1{s≤T0},1{t≤Tk}) tends to zero as k→∞ for all s, t ∈ [0,1). If,
in addition, EX
i <∞ for some γ > 1, then for any s, t ∈ [0,1) and k ∈ N,
we have
|cov(1{s≤T0},1{t≤Tk})| ≤ 2
γ(ρ(s) + ρ(t))k1−γ .(21)
Proof. The idea is to approximate 1{s≤T0} and 1{t≤Tk} by 1{s≤T (k/2)0 }
1{t≤T (k/2)
}, respectively; here by k/2 we mean ⌈k/2⌉, where ⌈x⌉=min{m ∈
Z :m≥ x}. Note that 1{s≤T (k/2)0 }
and 1{t≤T (k/2)
} are independent because the
first is a function of {Xi}i≤k/2 while the second is a function of {Xi}i≥k/2+1.
We then have
|cov(1{s≤T0},1{t≤Tk})|
= |cov(1{s≤T0},1{t≤Tk})− cov(1{s≤T (k/2)
},1{t≤T (k/2)
≤ |E(1{s≤T0}1{t≤Tk} − 1{s≤T (k/2)
}1{t≤T (k/2)
+ |E(1{s≤T0} − 1{s≤T (k/2)
})|+ |E(1{t≤Tk} − 1{t≤T (k/2)
})|(22)
= P{1{s≤T0}1{t≤Tk} 6= 1{s≤T (k/2)
}1{t≤T (k/2)
+ P{1{s≤T0} 6= 1{s≤T (k/2)0 }}+ P{1{t≤Tk} 6= 1{t≤T (k/2)k }
P{1{s≤T0}1{t≤Tk} 6= 1{s≤T (k/2)
}1{t≤T (k/2)
≤ P{1{s≤T0} 6= 1{s≤T (k/2)
} ∪ 1{t≤Tk} 6= 1{t≤T (k/2)
therefore the result follows from Lemma 1.
6. The r.v.’s {Ti}i∈Z, {T (k)i }i∈Z, and {Ti,n}
i=1 are associated ; the author
owes this observation to M. A. Lifshits.
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 17
Proof. Let us first recall the definition and some basic properties of as-
sociated variables. R.v.’s ξ1, . . . , ξm are associated if for any coordinate-wise
nondecreasing functions f, g :Rm →R, it is true that
cov(f(ξ1, . . . , ξm), g(ξ1, . . . , ξm))≥ 0
(assuming that the left-hand side is well defined). An infinite set of r.v.’s is
associated if any finite subset of its variables is associated.
The following sufficient conditions of association are well known; see [7].
(a) Independent variables are associated.
(b) Coordinate-wise nondecreasing functions (of finite number of argu-
ments) of associated r.v.’s are associated.
(c) If the variables ξ1,k, . . . , ξm,k are associated for every k and (ξ1,k, . . . ,
ξm,k)
D−→ (ξ1, . . . , ξm) as k→∞, then ξ1, . . . , ξm are associated.
(d) If two sets of associated variables are independent, then the union of
these sets is also associated.
Then {Ti,n}n−1i=1 are associated for every n by (a), (b) and (20). Analo-
gously, {T (k)i }i∈Z are associated for every k. Finally, since T
i → Ti a.s. as
k→∞ for every i, (c) ensures the association of {Ti}i∈Z.
7. For any s, t ∈R and k ∈ Z,
cov(1{T0≤s},1{Tk≤t})≥ 0.(23)
Proof. This inequality follows from cov(1{T0≤s},1{Tk≤t}) = cov(1{s<T0},
1{t<Tk}), the association of T0, Tk and (b).
8. If EX
i <∞ for some γ ≥ 2, then the stationary sequence min{Ti, t}
is strongly mixing for any t < 1 and its coefficients of strong mixing α(k)
satisfy α(k) = o(k2−γ).
Proof. Recall that stationary r.v.’s ξi are strongly mixing if α(k)→ 0 as
k→∞, where α(k) are the coefficients of strong mixing defined as
α(k) := sup
A∈F0−∞,B∈F
|P(AB)− P(A)P(B)|;
here F0−∞ := σ(ξ0, ξ−1, . . .) and F∞k := σ(ξk, ξk+1, . . .) are the σ-algebras of
“past” and “future,” respectively. It is readily seen that
α(k)≤ sup
0≤f,g≤1
| cov(f(ξ0, ξ−1, . . .), g(ξk, ξk+1, . . .))|,(24)
where the supremum is taken over Borel functions f, g :R∞ → [0,1].
Let us estimate α(k) in the same way we estimated the left-hand side
of (21). Fix some Borel functions f, g : R∞ → [0,1]. We approximate the
variables from the “past” T0∧ t, T−1∧ t, T−2∧ t, . . . by T (k/2)0 ∧ t, T
(k/2+1)
−1 ∧ t,
(k/2+2)
−2 ∧ t, . . . , respectively; and for the variables from the “future,” we
18 V. V. VYSOTSKY
use the analogous approximation. Now, f(T
(k/2)
0 ∧ t, T
(k/2+1)
−1 ∧ t, . . .) and
(k/2)
k ∧ t, T
(k/2+1)
k+1 ∧ t, . . .) are independent because the first is a function
of {Xi}i≤k/2 and the second is a function of {Xi}i≥k/2+1. We then argue in
the same way as in (22) to get
| cov(f(T0 ∧ t, T−1 ∧ t, . . .), g(Tk ∧ t, Tk+1 ∧ t, . . .))|
(T−i ∧ t) 6= (T (k/2+i)−i ∧ t)
(Tk+i ∧ t) 6= (T (k/2+i)k+i ∧ t)
i=k/2
P{(T0 ∧ t) 6= (T (i)0 ∧ t)}.
Now, by the formula of total probability, we have
P{(T0 ∧ t) 6= (T (i)0 ∧ t)}
= P{(T0 ∧ t) 6= (T (i)0 ∧ t), T
0 ≥ t}+ P{(T0 ∧ t) 6= (T
0 ∧ t), T
0 < t}
≤ P{1{t≤T0} 6= 1{t≤T (i)
}}+ P{T0 6= T
0 , T
0 ≤ t}
and combining all the estimates together, by Lemma 1 (24) and arbitrariness
of f and g, we get α(k)≤ 8
i=k/2 o(i
1−γ) = o(k2−γ) if γ > 2. For γ = 2, we
get α(k)≤ 16
i=k/2 P{inf i≥M Sii < t
2}= o(1) using the same argument and
applying (13), (14), and Fact 2 instead of Lemma 1.
3.5. The last collision. We finish this section with a statement on the
convergence of the moments of the last collision.
Proposition 3. In the i.d. model, T lastn
P−→ 1 as n→∞ if EX2i <∞.
This result is well known for the Poisson model; see Giraud [8].
Proof of Proposition 3. Let us first prove that P{T lastn ≥ t}→ 0 as
n→∞ for all t > 1. Since T lastn =max1≤j≤n−1Tj,n, we have
P{T lastn ≥ t} ≤
P{Tj,n ≥ t}.(25)
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 19
By taking into account that the minima in (17) are nonpositive and by
arguing as in (18),
P{Tj,n ≥ t} ≤ P
1≤k≤j∨n−j
(k− i)(Xj+i+1 − t2) +Xj+1 − t2 ≥ 0
1≤k≤j∨n−j
(k− i+1)(Xi − t2)≥ 0
1≤k≤n/2
(Si − it2)≥ 0
We claim that (without any assumptions on the moments of Xi)
P{Tj,n ≥ t} ≤ P
i≥(t−1)/4tn
1 + t2
;(26)
recall that t > 1. Clearly, (26) follows if we check that
1≤k≤n/2
(Si − it2)≥ 0
i≥(t−1)/4tn
1 + t2
Assume the converse; then, by the nonnegativity of Si,
(Si − it2) =
(Si − it2) +
i=cn+1
(Si − it2)
(Scn − it2) +
i=cn+1
1 + t2
− it2
where c := t−1
. We estimate the last expression with
cnScn −
(cn)2
t2 − (n/2)
2 − (cn)2
2 − 1
n2 − 1/4− c
2 − 1
It is simple to check that the right-hand side is negative, thus we have a
contradiction.
Then from (25), (26) and Fact 2 it follows that P{T lastn ≥ t} =
i=1 o((cn)
−1) = o(1) for all t > 1.
Now let us prove that P{T lastn < t} → 0 as n → ∞ for all t < 1. Since
T lastn =max1≤j≤n−1Tj,n, we estimate
P{T lastn < t} ≤ P
n,n < t
P{1{t≤Tj√n,n} 6= 1{t≤T (
20 V. V. VYSOTSKY
In view of (10) and Lemma 1, the sum is
j=1 o(n
−1/2) = o(1), hence it
remains to check that the first probability in the last line tends to zero.
For a fixed n, all T
are independent because each one is a function of
{Xi}|j√n−i|≤√n/2 (to be precise, of Xj√n−√n/2+2, . . . ,Xj√n+√n/2). Thus,
n−1{T (
n/2)√
< t} ≤ P
n−1{T0 < t},
which tends to zero; indeed, P{T0 < t}< 1 by Property 3, Section 3.4. �
4. Proofs of Fact 1 and Theorem 1 for the i.d. model. Recall that the
number of clusters Kn(t) is given by (6). Our idea is to study
i=1 1{t<Ti}
instead of
i=1 1{t<Ti,n}: We thus deal with a single sequence Ti and avoid
considering the triangular array Ti,n.
Let us now prove Fact 1 for the i.d. model. We prove (1) for t 6= 1 without
any additional assumptions on Xi; for t = 1, we require EX
i < ∞. The
properties of the limit function a(t) were studied in Section 3.4, Properties
3 and 4.
Proof of Fact 1. We put
a(t) := P{T0 > t}.(27)
Let us first prove (1) for all t < 1. It is sufficient to check that
Kn(t)
1{t<Ti}
P−→ 0, n→∞.(28)
Indeed, the stationary sequence 1{t<Ti} satisfies the law of large numbers by
Property 5, Section 3.4, and the well-known result of S. N. Bernstein:
Fact 3. The law of large numbers holds for r.v.’s ξi if there exists a
sequence r(k)→ 0 such that cov(ξi, ξj)≤ r(|i− j|) for all i, j ∈N.
By (6),
Kn(t)
1{t<Ti}
(1{t<Ti,n} − 1{t<Ti}),
where we used (10) to get the nonnegativity of the right-hand side. Then
(28) immediately follows from the Chebyshev inequality provided that the
expectation of the right-hand side tends to zero. By using (10), we obtain
E(1{t<Ti,n} − 1{t<Ti})≤
(E1{t<T (i∧n−i)
} − E1{t<Ti})
P{1{t<Ti} 6= 1{t<T (i∧n−i)
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 21
which is 2
i=1 o(1) = o(1) by Lemma 1. To be very precise, Lemma 1
deals with slightly different indicators, but we can estimate the considered
probability by repeating the proof of Lemma 1 word for word (or just use
Property 4, Section 3.4).
We now check that (1) holds for all t > 1. Using (26) gives E
Kn(t)
i=1 P{Ti,n > t} → 0 as n→∞ and
Kn(t)
P−→ a(t) = 0 follows from the
Chebyshev inequality.
It remains to check that (1) holds for t = 1 if EX2i < ∞ to conclude
the proof. If DXi = 0, then the situation is deterministic, this case was
described in Introduction. Here we always have Kn(1) = 1 and (1) is true.
If 0<DXi <∞, then by Property 3 from Section 3.4, we have a(1) = 0 and
P{T0 = 1} = 0; consequently, a(t) = P{T0 > t} is continuous at t= 1. Then
(1) is true for t = 1 since 0 <
Kn(1)
≤ Kn(t)
P−→ a(t) for any t ∈ (0,1) and
a(t)→ a(1) = 0 as tր 1. �
Now we prove Theorem 1 for the i.d. model. We think of D[0,1] as of a
separable metric space equipped with the Skorohod metric d, which induces
the Skorohod topology.
Proof of Theorem 1. At first, we prove (2). In view of representation
(6) for Kn(t), relation (2) follows from the relation
0≤t≤1−ε
1{t<Ti,n} −
1{t<Ti}
P−→ 0 for all ε ∈ (0,1)(29)
and the existence of a centered Gaussian process K(·) on [0,1) such that
1{t<Ti} −na(t)
D−→K(·) in D[0,1− ε] for all ε ∈ (0,1).(30)
Indeed, if Yn
D−→ Y and d(Yn, Y ′n)
P−→ 0 for some random elements Yn, Y ′n, Y
of the separable metric spaceD[0,1−ε], then Y ′n
D−→ Y ; recall that d(Yn, Y ′n)≤
supt∈[0,1−ε] |Yn(t)− Y ′n(t)|.
We start with (29). It is sufficient to prove that the expectation of the
left-hand side tends to zero. Since the supremum of a sum does not exceed
the sum of suprema, let us check that
E sup
0≤t≤1−ε
|1{t<Ti,n} − 1{t<Ti}| −→ 0 for all ε ∈ (0,1).(31)
By (10), we have
E sup
0≤t≤1−ε
|1{t<Ti,n} − 1{t<Ti}| ≤ E sup
0≤t≤1−ε
(1{t<T (i∧n−i)
} − 1{t<Ti})
22 V. V. VYSOTSKY
= P{Ti 6= T (i∧n−i)i , Ti ≤ 1− ε}
= P{Ti 6= T (i∧n−i)i , T
(i∧n−i)
i < 1− ε}
+ P{1{1−ε≤Ti} 6= 1{1−ε≤T (i∧n−i)
where the last equality was obtained via the formula of total probability.
Combining the estimates together and using Lemma 1,
E sup
0≤t≤1−ε
|1{t<Ti,n} − 1{t<Ti}|
≤ 2ρ(1− ε)√
(i ∧ n− i)1−γ = 4ρ(1− ε)√
i1−γ .
The last expression is O(n3/2−γ) and (31), which implies (29), follows.
Now let us prove (30). As long as
Un(t) :=−
1{t<Ti} − na(t)
1{Ti≤t} − (1− a(t))
the Un(·) is the empirical process of stationary r.v.’s Ti with the continuous
common distribution function 1− a(t). By K(·) D=−K(·), (30) is equivalent
to the existence of a centered Gaussian process K(·) on [0,1) such that
Un(·)
D−→K(·) in D[0,1− ε] for all ε ∈ (0,1).(32)
We will use the following result from Lin and Lu [17], Section 12 on
convergence of empirical processes. They attribute this statement to Q.-M.
Shao, who published it in 1986, in Chinese.
Fact 4. Let ξi be a sequence of stationary strongly mixing r.v.’s dis-
tributed on [0,1], and let F be the common distribution function of ξi.
Suppose F (x) = x on [0,1] (i.e., ξi are uniformly distributed) and the co-
efficients of strong mixing of the sequence F (ξi) decrease as O(k
−(2+δ))
as k → ∞ for some δ > 0. Then the empirical processes of ξi weakly con-
verge in D[0,1] to a centered Gaussian process with the covariance function
i∈Z cov(1{ξ0≤s},1{ξi≤t}).
Remark. The limit Gaussian process is a.s. continuous on [0,1]. Fact 4
also holds true if F is an arbitrary continuous distribution function.
The a.s. continuity of the limit process could be concluded by a compar-
ison of the proof from Lin and Lu [17] with the proof of Theorem 22.1 from
Billingsley [3]. The statements and the proofs of these theorems are identical,
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 23
but Lin and Lu do not state the continuity while Billingsley does. Further,
since F (ξi) is uniformly distributed on [0,1] if F is continuous, Fact 4 holds
true for every continuous F ; see the proof of Theorem 22.1 by Billingsley [3]
for explanations.
Recall that we need to prove the convergence of the empirical process of
Ti. It seems that the r.v.’s Ti are not strongly mixing; but min{Ti,1− ε} are
strongly mixing because of Property 8, Section 3.4. These variables are not
continuous and so we need to fix them. Let us fix an ε ∈ (0,1), and let αi
be i.i.d. r.v.’s independent of all Ti and, say, uniformly distributed on [0, ε];
we define T̃i := min{Ti,1− ε}+ 1{Ti≥1−ε}αi.
The stationary variables T̃i are distributed on [0,1], their common dis-
tribution function G is continuous, and the coefficients of strong mixing
of G(T̃i) decrease as o(k
2−γ). The proof of the last statement is the same
as the proof of Property 8 from Section 3.4. Indeed, approximate the vari-
ables G(T̃0),G(T̃−1), . . . from the “past” by G(T̃
(k/2)
0 ),G(T̃
(k/2+1)
−1 ), . . . where
i := min{T
i ,1− ε}+ 1{T (m)
≥1−ε}αi; use the analogous approximation
for the variables from the “future”; and then repeat word for word the ar-
guments of the previous proof.
Now, recalling that γ > 4, we see that T̃i satisfy the assumptions of Fact 4,
with the only difference that their distribution is not uniform. By Ũn(·)
denote the empirical process of T̃i; clearly, Ũn(·) coincides with the empirical
process Un(·) of Ti on [0,1− ε]. By the remark to Fact 4, we conclude that
first,
Ũn(·)
D−→ K̃(·) in D[0,1],(33)
where K̃(·) is a centered Gaussian process with the covariance function
R̃(s, t) :=
cov(1{T̃0≤s},1{T̃i≤t})
and, second, trajectories of K̃(·) are a.s. continuous on [0,1].
[There exists a simpler and more elegant proof of (33). Note that {T̃i}i∈Z
are associated as coordinate-wise nondecreasing functions of associated r.v.’s
{Ti, αi}i∈Z, see (a), (b) and (d) from Property 6, Section 3.4. Then we can
obtain (33) applying the result of Louhichi [18] on convergence of empirical
processes of stationary associated r.v.’s ξi instead of using Fact 4. This the-
orem requires only cov(F (ξ0), F (ξk)) = O(k
−(4+δ)), which could be proved
analogously to Property 5, Section 3.4. Thus we avoid the complicated esti-
mations of the strong mixing coefficients, and the proof of (33) is becomes
much simpler. The only problem is that this proof requires γ > 5.
We also note that the a.s. continuity of K̃(·) could be proved directly,
without referring to the proof of Fact 4. The arguments should be the same
as in the proof of the continuity of KUnif(·) in Section 5.]
24 V. V. VYSOTSKY
Define
R(s, t) :=
cov(1{T0≤s},1{Ti≤t}),(34)
which is, evidently, equal to R̃(s, t) on [0,1 − ε]2. Since R̃(s, t) is positive
definite and ε > 0 is arbitrary, the function R(s, t) is positive definite on
[0,1)2. Therefore, by Lifshits [15], Section 4, there exists a centered Gaussian
process K(·) on [0,1) with the covariance function R(s, t). The trajectories
of K(·) are a.s. continuous on [0,1) by K(·) D= K̃(·) on [0,1−ε], arbitrariness
of ε > 0, and the a.s. continuity of K̃(·) on [0,1].
Finally, by (33), Ũn(·) = Un(·) on [0,1− ε], K̃(·) D=K(·) on [0,1− ε], and
the a.s. continuity of K̃(·), we get (32). Since (32) implies (30), we conclude
the proof of (2).
Only the stated properties of R(s, t) remain to be proven. The continuity
of the joint distribution function of continuous variables T0 and Ti implies
that cov(1{T0≤s},1{Ti≤t}) is continuous on [0,1)
2 for every i ≥ 0. Then, in
view of (21), R(s, t) is continuous on [0,1)2 as a sum of uniformly converging
series of continuous functions.
The strict positivity of R(s, t) on (
µ,1)2 trivially follows from (34), (23)
and cov(1{T0≤s},1{T0≤t}) = a(s∨ t)(1−a(s∧ t)) > 0; the last inequality holds
by Property 3, Section 3.4. The R(s, t) = 0 on [0,1)2 \ (√µ,1)2 follows from
P{Ti ≤
µ}= 0, which holds by Properties 3 and 4 from Section 3.4. �
We note that (3) holds for t 6= 1 under the less restrictive condition EX2i <
∞. For t < 1, the proof is almost the same: By (29), which is true for γ > 3/2,
we conclude that (3) holds if the stationary associated sequence 1{t<Ti}
satisfies the central limit theorem. Then we refer to the central limit theorem
for stationary associated sequences from Newman [21]; his theorem requires
only R(t, t)<∞, that is, the convergence of the right-hand side of (34). This
condition holds by (13) and Fact 2. For t > 1, relation (3) holds true with
σ2(t) = 0 because of Proposition 3.
Finally, note that the process K(·) is associated, that is, the r.v.’s
{K(t)}t∈[0,1) are associated. In fact, by (6), Property 6 from Section 3.4,
and Condition (b) from the same Property 6, the processes
Kn(·)−na(·)√
associated for every n. Then K(·) is associated by (2) and (c), Property 6.
5. Proof of Theorem 1 for the uniform model. There exists a simple
method that allows to extend results from the Poisson model to the uniform
model and vise versa. The method is based on the next statement (see
Karlin [13], Section 9.1).
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 25
Fact 5. Let Si be an exponential random walk. Then for any k ≥ 1, we
, . . . ,
= (U1,k,U2,k, . . . ,Uk,k),(35)
where Ui,k are the order statistics of k i.i.d. r.v.’s uniformly distributed on
[0,1]. Moreover, the random vector in the left-hand side of (35) is indepen-
dent of Sk+1.
Therefore, if xPoissj,n (0) =
Sj are the initial positions of particles in the
Poisson model, then for the initial positions of particles in the uniformmodel,
we have xUnifj,n (0) =
· xPoissj,n (0). By Proposition 2 and (5), we conclude
TUnifj,n = β
Poiss
j,n , βn :=
,(36)
and hence, using (6), we get
KUnifn (t) =K
Poiss
n (βnt).(37)
Note that the process KUnifn (·) and the r.v. βn are independent since val-
ues of the process are defined by xUnif1,n (0), . . . , x
n,n (0), which are mutually
independent of βn by Fact 5.
Now we prove Theorem 1 for the uniform model.
Proof of Theorem 1. Denote
Yn(t) :=
KUnifn (t)− na(t)√
, Zn(t) :=
n(a(t)− a(βnt));
we stress that Yn(·) and Zn(·) are independent.
Fix an ε ∈ (0,1). First, it follows from (2) for the Poisson model and (37)
Yn(·) +Zn(·)
D−→KPoiss(·) in D[0,1− ε].(38)
Indeed, the process Yn(·) +Zn(·) is obtained from 1√n(K
Poiss
n (·)− na(·)) by
the random time change t 7→ βnt; and since ‖βnt− t‖C[0,1−ε]
P−→ 0, we have
Yn(·) +Zn(·),
KPoissn (·)− na(·)√
P−→ 0
by the definition of the Skorohod metric d.
Second, from Fact 1, (15), and (27) it follows that aUnif(t) = aPoiss(t) =
P{TPoiss0 ≥ t}= 1− t2 for 0≤ t≤ 1, and by the central limit theorem,
Zn(t)
D−→ t2η in D[0,1− ε],(39)
26 V. V. VYSOTSKY
where η is a standard Gaussian r.v.
We claim that (38), the independence of Yn(·) and Zn(·), and (39) yield
the weak convergence of Yn(·) in D[0,1− ε]. Let us check the tightness of
Yn(·) and the convergence of their finite-dimensional distributions.
The tightness of Yn(·) in D[0,1− ε] follows from Yn(·) = (Yn(·)+Zn(·))−
Zn(·), (38), and (39). Indeed, by the Prokhorov theorem, (38) and (39) yield
that both sequences Yn(·) + Zn(·) and −Zn(·) are tight. But trajectories
of −Zn(·) are a.s. continuous because of the continuity of a(·), and the
tightness follows from the continuity of addition + :D×C →D and the fact
that under any continuous mapping, the image of a compact set is also a
compact set.
Now we study convergence of finite dimensional distributions of Yn(·).
Recall that the characteristic function of a centered Gaussian vector in Rm
is e−1/2(Ru,u), where u ∈ Rm and R is the covariance matrix of the vector.
Then (38), the independence of Yn(·) and Zn(·), and (39) yield that for the
characteristic functions of all finite-dimensional distributions of Yn(·), we
Eei(Yn(t),u) −→ e−1/2({R
Poiss(tj ,tk)−t2j t
j,k=1
,(40)
where u ∈Rm, t= (t1, . . . , tm) ∈ [0,1−ε]m, and Yn(t) := (Yn(t1), . . . , Yn(tm)).
We stress that (40) is true for every t ∈ [0,1− ε]m since the limit processes
in (38) and (39) have continuous trajectories.
We see that the matrix {RPoiss(tj , tk)− t2j t2k}mj,k=1 is positive definite for
any t = (t1, . . . , tm) ∈ [0,1− ε]m and m≥ 1 since the absolute value of the
left-hand side of (40) does not exceed one. Putting
RUnif(s, t) :=RPoiss(s, t)− s2t2,
we have {RPoiss(tj , tk)− t2j t2k}mj,k=1 = {RUnif(tj , tk)}mj,k=1; then the function
RUnif(s, t) is positive definite on [0,1)2 since ε > 0 is arbitrary. Thus, by
Lifshits [15], Section 4, RUnif(s, t) is the covariance function of some centered
Gaussian process KUnif(·) on [0,1).
Relation (2) is thus proved. Now check that KUnif(·) ∈C[0,1− ε] a.s. to
conclude the proof of Theorem 1 for the uniform model.
For this purpose, let us prove that a.s., trajectories of Yn(·) have jumps
of size 1√
only. In fact, the jumps of Yn(·) coincide with the jumps of
KUnifn (·), whose jumps are of size 1√n if and only if T
6= TUnifj2,n for
1 ≤ j1 6= j2 ≤ n − 1. By (36), we need to verify that TPoissj1,n 6= T
Poiss
for 1 ≤ j1 6= j2 ≤ n − 1. This relation follows from (20) if H(k1, j1, l1) 6=
H(k2, j2, l2) a.s. for j1 6= j2 and k1, k2, l1, l2 ≥ 1. The last a.s. nonequality is
obvious because if the equality holds true, then a certain nontrivial linear
combination of i.i.d. exponential Xi equals zero.
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 27
Then there exist a.s. continuous Ỹn(·) such that supt∈[0,1−ε] |Ỹn(t)−Yn(t)| ≤
a.s.; consequently, d(Ỹn, Yn) ≤ 1√n a.s. Then by Yn(·)
D−→ KUnif(·), we
also have Ỹn(·)
D−→KUnif(·). But 1 = lim inf P{Ỹn(·) ∈C} ≤ P{KUnif(·) ∈C}
since C ⊂D is closed in the Skorohod topology, therefore, a.s., KUnif(·) is
continuous on [0,1− ε].
Since ε ∈ (0,1) is arbitrary, a.s., KUnif(·) is continuous on the whole inter-
val [0,1). The RUnif(s, t) =RPoiss(s, t)− s2t2 is continuous on [0,1)2 because
RPoiss(s, t) is. �
6. The number of clusters at the critical moment. Now we turn our
attention to the number of clusters at the critical moment t = 1. We are
interested in the behavior of
Kn(1)− na(1)√
Kn(1)√
which is the left-hand side of (3) at t = 1; here we have a(1) = 0 under
EX2i <∞, see Property 3, Section 3.4.
We do not know if this sequence is weakly convergent, but we hope that
it is. We also have a naive guess that its limit is Gaussian because the limit
in Theorem 1 is Gaussian. In view of Kn(1)≥ 1, this conjectured weak limit
is nonnegative, hence it is Gaussian if and only if it is identically equal to
zero. However, the results of this section show that the limit is nonzero, thus
our guess on Gaussianity fails.
The study of convergence of
Kn(1)√
is quite complicated. Therefore, in this
section, we consider only the Poisson model. First, let us prove the following
statement.
Proposition 4. In the Poisson model, we have limn→∞P{Kn(1) =
1}> 0.
Proof. On the one hand, Kn(1) = 1 is equivalent to T
n;Poiss ≤ 1, where
T lastn;Poiss denotes the moment of the last collision in the Poisson model. On
the other hand, a result by Giraud [8] states that in the uniform model,
n(T lastn;Unif − 1)
D−→ sup
0≤x≤1
W (y)dy −
W (y)dy
=: τ,
where
W (·) is a Brownian bridge. Now, by (36), we have T lastn;Unif = β−1n T lastn;Poiss,
hence
n(β−1n T
n;Poiss − 1)
D−→ τ.(41)
28 V. V. VYSOTSKY
But from the central limit theorem and the law of large numbers,
n(β−1n − 1) =−
Sn+1 − n√
Sn+1(
Sn+1 +
D−→ η
,(42)
where η is a standard Gaussian r.v. and Si is a standard exponential ran-
dom walk that defines initial positions of particles. Since, in view of Fact 5,
T lastn;Unif = β
n;Poiss and βn are independent, from (41), (42), and the law
of large numbers it follows that
n(T lastn;Poiss − 1)
D−→ τ − η
= τ +
where τ and η are independent. Thus,
P{Kn(1) = 1}= lim
P{T lastn;Poiss ≤ 1}= P
The main advantage of the Poisson model is that, by Lemma 2 and Prop-
erty 4, Section 3.4 we have P{Tj,n > 1}= epjpn−j , where
pk := P
1≤m≤k
(Si − ESi)≥ 0
and Si is a standard exponential random walk. We say that the sequence of
r.v.’s
i=1(Si−ESi) is an integrated random walk. In the proof of Property
3, Section 3.4, we showed that pk → 0 as k→∞. Therefore, it is reasonable
to say that pk are the unilateral small deviation probabilities of an integrated
centered random walk.
We need to obtain the asymptotics of pk → 0 to continue the study of
convergence of
Kn(1)√
. Unfortunately, the results of the rest of this section
are completely dependent on the correctness of the following conjecture.
Conjecture 1. We have pk ∼ c1k−1/4 as k→∞ for some c1 ∈ (0,∞).
Simulations show that the conjecture is true and c1 ≈ 0.36. The weaker
form pk ≍ k−1/4 of Conjecture 1 was proved by Sinai [22], but only for inte-
grated symmetric Bernoulli random walks. It also interesting to note that,
by McKean [19], the unilateral small deviation probabilities of an integrated
Wiener process have the same order as T →∞:
0≤s≤T
W (u)du≥−1
∼ c2T−1/4(43)
for some c2 ∈ (0,∞). The left-hand side of (43) is a unilateral small deviation
probability since
0≤s≤T
W (u)du≥−1
0≤s≤1
W (u)du≥−T−3/2
.(44)
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 29
To be precise, McKean was interested in a more general problem, and
some calculations are required to obtain (43) from his results. Therefore,
we additionally refer to Isozaki and Watanabe [12] who state (43) explic-
itly.
By the results mentioned above, we also suppose that Conjecture 1 is
true for other integrated centered random walks that satisfy some moment
conditions.
Now we are able to prove the following result on convergence of
Kn(1)√
Proposition 5. Suppose Conjecture 1 holds true. Then in the Poisson
model, we have
Kn(1)√
= c3, sup
Kn(1)√
<∞(45)
for some c3 ∈ (0,∞); the sequence Kn(1)√n is tight and uniformly integrable;
and the limit of any weakly converging subsequence of
Kn(1)√
takes value zero
with positive probability, but is not identically equal to zero.
Numerical simulations show that
Kn(1)√
is weakly convergent and that this
convergence is quite fast. In Figure 1 we present the (empirical) distribution
function of
Kn(1)√
for n = 10,000. Since the simulations performed for n =
40,000 showed a hardly perceptible difference, this function seems to be a
good candidate for the distribution function of the conjectured limit.
Note that if we weaken Conjecture 1 to pk ≍ k−1/4, then Proposition 5
still holds true with the only difference that E
Kn(1)√
Proof of Proposition 5. We start with the convergence of the ex-
pectation. On the one hand, by (6) and Lemma 2,
Kn(1)√
pipn−i,
and on the other hand,
i−1/4(n− i)−1/4 = 1
)−1/4(
)−1/4
−→B(3/4,3/4)
as the integral sum of Beta function. Then it follows from Conjecture 1 and
standard arguments that E
Kn(1)√
converges to c3 := ec
1B(3/4,3/4) > 0.
30 V. V. VYSOTSKY
Fig. 1. The distribution function of
Kn(1)√
for n= 10,000.
Now we check the uniform boundedness of E(
Kn(1)√
)2. By (6) it is sufficient
to prove that
i,j=1,i 6=j
P{Ti,n > 1, Tj,n > 1}<∞.(46)
Suppose i < j; then by using (8) and properties of Fk,j,l(·), we get
P{Ti,n > 1, Tj,n > 1}= P
1≤k≤n−i
1≤l≤i
Fk,i,l(1)> 0, min
1≤k≤n−j
1≤l≤j
Fk,j,l(1)> 0
1≤k≤(j−i)/2
1≤l≤i
Fk,i,l(1)> 0, min
1≤k≤n−j
1≤l≤(j−i)/2
Fk,j,l(1)> 0
where by (j − i)/2 we mean ⌈(j − i)/2⌉. The minima in the last expres-
sion are independent as functions of {Xm}m≤(i+j)/2 and {Xm}m≥(i+j)/2+1,
respectively; hence
P{Ti,n > 1, Tj,n > 1} ≤ P
1≤k≤(j−i)/2
1≤l≤i
Fk,i,l(1)> 0
1≤k≤n−j
1≤l≤(j−i)/2
Fk,j,l(1)> 0
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 31
= P{Ti,i+(j−i)/2 > 1} · P{T(j−i)/2,n−j+(j−i)/2 > 1}
= e2pip
⌈(j−i)/2⌉pn−j,
where the first equality follows from (8) and the second follows from Lemma 2.
Recalling Conjecture 1, we get
i,j=1,i 6=j
P{Ti,n > 1, Tj,n > 1} ≤
i,j=1,i 6=j
e2pip
⌈|j−i|/2⌉pn−j
i,j=1,i 6=j
i−1/4⌈|j − i|/2⌉−1/2(n− j)−1/4
i,j=1,i 6=j
)−1/4∣
−1/2(
)−1/4
for some c > 0. The last expression is an integral sum converging to
x−1/4|x− y|−1/2(1− y)−1/4 dxdy,
and it is a simple exercise to check that the integral is finite. This concludes
(46).
The uniform integrability of
Kn(1)√
follows from the second relation from
(45), see Billingsley [3], Section 5, and the tightness follows from the uniform
integrability.
Finally, suppose
Kni(1)√
D−→ ξ for some subsequence ni → ∞ and some
r.v. ξ. Then Eξ = c3 > 0 by the uniform integrability and (45), and hence
ξ is not identically equal to zero. But the distribution of ξ has an atom at
zero since by Proposition 4 and properties of weak convergence,
P{ξ = 0}= lim
P{ξ ≤ ε}
≥ lim
lim sup
Kni(1)√
≥ lim
P{Kni(1) = 1}> 0. �
7. Open questions. 1. The number of clusters at the critical moment
t= 1.
Here the main question is if Conjecture 1 holds true. Even by itself, this
problem is worth studying.
But even if Conjecture 1 is true, we still do not have a proof of weak
convergence of
Kn(1)√
, it is only known that this sequence is tight. The author
32 V. V. VYSOTSKY
strongly believes, relying on numerical simulations, that the limit exists. It
would be interesting to find this conjectured limit, which should be nontrivial
by Proposition 5, in an explicit form.
2. The weak convergence of
Kn(·)−na(·)√
on the whole interval [0,1].
It is very natural to ask if it is possible to strengthen Theorem 1 by
proving the weak convergence of
Kn(·)−na(·)√
in D[0,1]. This complicated
problem returns us again to Question 1 because the weak convergence of
Kn(·)−na(·)√
in D[0,1] implies the weak convergence of
Kn(1)−na(1)√
Kn(1)√
, see
Billingsley [3], Section 15. But even if
Kn(1)√
converges, its weak limit K(1)
is not Gaussian, hence the limit process K(·), which is Gaussian on [0,1),
is no more Gaussian on [0,1]. Therefore, it is doubtful that Theorem 1 is
true in D[0,1]; at least, one should provide a proof completely different from
the presented one. Also, it is unclear how to define the finite-dimensional
distributions of the non-Gaussian K(·) on [0,1] because simulations show
that K(1) would not be independent with K(t) for t < 1.
3. The number of clusters in the warm gas.
In the presented case, initial speeds of particles are zero. This model is of-
ten called the cold gas according to its zero initial temperature. We introduce
a new model stating that initial speeds of particles are anv1, anv2, . . . , anvn,
where vi are some i.i.d. r.v.’s and an is a sequence of normalization con-
stants. This model, called the warm gas, was studied in many papers, for
example, [14, 16, 20, 25].
It is of a great interest to study the behavior of Kn(t) in the warm gas.
In [25], the author proved that in the basic case where an = 1 for all n and
Ev2i <∞, we have
Kn(t)
P−→ 0 for all t > 0. The question is to find a normal-
ization of Kn(t) leading to some nontrivial limit. Clearly, this normalization
depends on an, but it is very possible that there is an effect of phase tran-
sition similar to the one discovered by Lifshits and Shi [16]: If an are small
enough, then the gas has a low temperature and the normalization is the
same as in the cold gas. If an are big enough, as in the basic case an ≡ 1,
then the normalization and the behavior of the gas differ entirely from the
case of the cold gas.
The author believes that the localization property, which is described in
Section 3, could be helpful in a study of these questions.
It is also interesting to compare the behavior of Kn(1) in the warm and in
the cold gases; in the warm gas, the moment t= 1 plays the same “critical”
role as in the cold gas, see Lifshits and Shi [16]. The variable Kn(1) was
studied by Suidan [24], who considered the warm gas with an ≡ 1 and deter-
ministic initial positions of particles (his initial positions were 1
, . . . , n
For this case, Suidan found the distribution of Kn(1) and showed that
EKn(1)∼ logn. Recall that in the presented case, EKn(1)∼ c3
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 33
4. The number of clusters in ballistic systems of sticky particles.
A sticky particles model is called ballistic if it evolves according to the
laws introduced in Section 1, but in the absence of gravitation. Such models
are, in some sense, more natural than gravitational ones because the ba-
sic assumption that gravitation does not depend on distance is sometimes
confusing. However, an unpublished paper of Lifshits and Kuoza shows that
certain gravitational and ballistic models are tightly connected.
It seems interesting to study the number of clusters in the ballistic model.
The author does not know any results in this field.
Acknowledgments. I am grateful to my adviser Mikhail A. Lifshits for
drawing my attention into the subject and for his guidance. I also thank the
anonymous referees for carefully reading this paper and useful comments.
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[20] Martin, Ph. A. and Piasecki, J. (1996). Aggregation dynamics in a self-gravitating
one-dimensional gas. J. Statist. Phys. 84 837–857. MR1400187
[21] Newman, C. M. (1980). Normal fluctuations and the FKG inequalities. Comm.
Math. Phys. 74 119–128. MR0576267
[22] Sinai, Ya. G. (1992). Distribution of some functionals of the integral of a random
walk. Theor. Math. Phys. 90 219–241. MR1182301
[23] Shandarin, S. F. and Zeldovich, Ya. B. (1989). The large-scale structure of the
universe: Turbulence, intermittency, structures in a self-gravitating medium.
Rev. Modern Phys. 61 185–220. MR0989562
[24] Suidan, T. M. (2001). A one-dimensional gravitationally interacting gas and the con-
vex minorant of Brownian motion. Russ. Math. Surv. 56 687–708. MR1861441
[25] Vysotsky, V. V. (2006). On energy and clusters in stochastic systems of sticky
gravitating particles. Theory Probab. Appl. 50 265–283. MR2221711
[26] Vysotsky, V. V. (2007). The area of exponential random walk and partial sums of
uniform order statistics. J. Math. Sci. 147 6873–6883.
Department of Probability Theory
and Mathematical Statistics
Faculty of Mathematics and Mechanics
St. Petersburg State University
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Stary Peterhof 198504
Russia
E-mail: vlad.vysotsky@gmail.com
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	Introduction
	Description of the model
	Statement of the problem and the results
	Organization of the paper
	Method of barycenters
	Study of the i.d. model. The localization property
	The initial study
	Localization property of the aggregation process
	The distribution function of T0 in the Poisson model
	Some properties of the variables Ti
	The last collision
	Proofs of Fact 1 and Theorem 1 for the i.d. model
	Proof of Theorem 1 for the uniform model
	The number of clusters at the critical moment
	Open questions
	Acknowledgments
	References
	Author's addresses

  We give a quantitative analysis of clustering in a stochastic model of
one-dimensional gas. At time zero, the gas consists of $n$ identical particles
that are randomly distributed on the real line and have zero initial speeds.
Particles begin to move under the forces of mutual attraction. When particles
collide, they stick together forming a new particle, called cluster, whose mass
and speed are defined by the laws of conservation. We are interested in the
asymptotic behavior of $K_n(t)$ as $n\to \infty$, where $K_n(t)$ denotes the
number of clusters at time $t$ in the system with $n$ initial particles. Our
main result is a functional limit theorem for $K_n(t)$. Its proof is based on
the discovered localization property of the aggregation process, which states
that the behavior of each particle is essentially defined by the motion of
neighbor particles.

Introduction.
1.1. Description of the model. We give a quantitative analysis of clus-
tering in a stochastic model of one-dimensional gas. At time zero, the gas
consists of n point particles, each one of mass 1
. These particles are ran-
domly distributed on the real line and have zero initial speeds. Particles
begin to move under the forces of mutual attraction. When two or more
particles collide, they stick together forming a new particle, called cluster,
whose mass and speed are defined by the laws of mass and momentum
conservation. Between collisions, particles move according to the laws of
Newtonian mechanics.
We suppose that the force of mutual attraction does not depend on dis-
tance and equals the product of masses. This assumption is natural for
Received March 2007; revised September 2007.
1Supported in part by the Grants NSh-4222.2006.1 and DFG-RFBR 436 RUS
113/773/0-1(R).
AMS 2000 subject classifications. Primary 60K35, 82C22; secondary 60F17, 70F99.
Key words and phrases. Sticky particles, particle systems, gravitating particles, number
of clusters, aggregation, adhesion.
This is an electronic reprint of the original article published by the
Institute of Mathematical Statistics in The Annals of Applied Probability,
2008, Vol. 18, No. 3, 1026–1058. This reprint differs from the original in
pagination and typographic detail.
http://arxiv.org/abs/0704.0086v2
http://www.imstat.org/aap/
http://dx.doi.org/10.1214/07-AAP481
http://www.imstat.org
http://www.ams.org/msc/
http://www.imstat.org
http://www.imstat.org/aap/
http://dx.doi.org/10.1214/07-AAP481
2 V. V. VYSOTSKY
one-dimensional models because, by the Gauss law applied to flux of the
gravitational field, gravitation is proportional to the distance to the power
one minus dimension of the space. At any moment, the acceleration of a
particle is thus equal to difference of masses located to the right and to the
left of the particle.
Random initial positions of particles are usually described (see [8, 16,
25]) by the following natural models: in the uniform model, n particles are
independently and uniformly spread on [0,1]; in the Poisson model, particles
are located at points 1
S2, . . . ,
Sn, where Si is a standard exponential
random walk. In other words, particles are located at points of first n jumps
of a Poisson process with intensity n.
These two models are the most natural and interesting; let us call them
the main models of initial positions. However, we will see that behavior
of the Poisson model is essentially defined by independence of initial dis-
tances between particles rather than by the particular type of the distances’
distribution. Therefore, it is of a great mathematical interest to general-
ize the Poisson model by introducing the i.d. model, where “i.d.” stands
for “independent distances,” as follows. Particles are initially located at
S2, . . . ,
Sn, where Si is a positive random walk whose nonnegative
i.i.d. increments Xi satisfy the normalization condition EXi = 1. Note that
if we proceed to the limit as n→∞, we consider a system of total mass one,
which consists of, roughly speaking, infinitesimal particles homogeneously
spread on [0,1]; this is true for all the mentioned models of initial positions.
The mathematical interest in sticky particles systems arises mainly from
relations between these systems and some nonlinear partial differential equa-
tions originating from fluid mechanics, for example, the Burgers equation.
These equations admit interpretation in terms of sticky particles; see Gur-
batov et al. [10], Brenier and Grenier [4] or E, Rykov and Sinai [6]. Sticky
particles models are also used for numerical solving of other partial differen-
tial equations; see Chertock et al. [5] for explanations and further references.
As time goes, particles aggregate in clusters. Clusters become larger and
larger while the number of clusters decreases until they merge into a single
cluster containing all initial particles. This process of mass aggregation is
strongly connected with additive coalescence; see Bertoin [2] and Giraud [9]
for the most recent results and references.
The aggregation process resembles formation of a star from dispersed
space dust and sticky particles models indeed have relations to astrophysics.
It is appropriate to clarify these relations since they are not so direct and
cause a lot of misunderstanding.
It is known that the distribution of galaxies in the universe is very inho-
mogeneous and the regions of high density form a peculiar cellular structure.
The first attempt to understand the formation of such structures was made
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 3
in 1970 by Zeldovich. Most of the mass in the universe is believed to ex-
ist in the form of particles that practically do not collide with each other
and interact only gravitationally, for example, neutrinos. In his model, Zel-
dovich considered an initially homogeneous collisionless medium of particles
moving by pure inertia; the gravitational interaction was taken away by an
appropriate time change. He showed that singularities, that is, the thin re-
gions of very high density of particles, so called “pancakes,” appear even if
initial speeds of particles form a smooth velocity field.
Zeldovich’s approximate model, however, does not explain formation of
the cellular structure of matter. His approximation does not take into ac-
count that particles hitting a “pancake” are hampered by its strong gravita-
tional field and start oscillating inside the “pancake” instead of flying away.
Although this gravitational adhesion of collisionless particles is not precisely
the same as the real sticking, the model of sticky particles serves as a reason-
able approximation. The effect of gravitational adhesion was then analyzed
by the use of the Burgers equation; Gurbatov, Saichev and Shandarin pro-
posed it in 1984 to extend Zeldovich’s approximation, which is invalid after
formation of “pancakes.”
The model of sticky particles is directly mentioned in Gurbatov et al. [11];
a comprehensive survey of the formation of the Universe’s large-scale struc-
ture could be found in Shandarin and Zeldovich [23].
1.2. Statement of the problem and the results. In general, the problem
is to describe the process of mass aggregation. How fast is it? How large
the clusters are? Where do clusters appear most intensively, and so forth?
Numerous papers on the model (e.g., [8, 14, 16, 20, 25]) are dedicated to
probabilistic description of various properties of the aggregation process as
the number of initial particles n tends to infinity. Thus, the behavior of a
typical system consisting of a large number of particles is studied.
In this paper, we are interested in the asymptotic behavior of Kn(t), which
denotes the number of clusters at time t in the system with n initial particles.
This variable is a decreasing random step function satisfying Kn(0) = n
and Kn(t) = 1 for t ≥ T lastn , where T lastn denotes the moment of the last
collision. While calculating Kn(t), we also count initial particles that have
not experienced any collisions; in other words, Kn(t) is the total number of
particles existing at time t.
It is very important to know the behavior of Kn(t). This gives us a deep
understanding of the aggregation process since the average size of a cluster
at time t is n
Kn(t)
At first we give a short deterministic example. Suppose that particles are
located at points 1
, . . . , n
, that is, Si = i. By simple calculations, we find
that there would not be any collisions before t= 1. At the moment t= 1, all
4 V. V. VYSOTSKY
particles simultaneously stick together, hence Kn(t) = n for 0 ≤ t < 1 and
Kn(t) = 1 for t≥ 1.
However, when the initial positions are random, the aggregation process
behaves entirely differently. In [25], the author proved the following state-
ment.
Fact 1. There exists a deterministic function a(t) such that both in the
Poisson and the uniform models of initial positions, for any t≥ 0, we have
Kn(t)
P−→ a(t), n→∞.(1)
The function a(t) is continuous, a(0) = 1, and a(t) = 0 for t≥ 1. We conjec-
ture, on the basis of numerical simulations, that a(t) = 1− t2 for 0≤ t≤ 1.
The relation a(t) = 0 for t > 1 is not of a surprise because we know from
Giraud [8] that both in the Poisson and the uniform models, T lastn
P−→ 1 (the
limit constant is so “fine” due to the proper scaling of the model). Therefore,
we say that the moment t= 1 is critical ; note that this moment coincides
with the moment of the total collision in the deterministic model.
The aim of this paper is to strengthen the result of [25]. We first gen-
eralize Fact 1 and prove it for the i.d. model. We will see [relations (19)
and (27) below] that a(t) is equal to the probability of a certain event that
is expressed in terms of Xi. Also, we will prove that a(t) depends on the
common distribution of Xi as follows: a(t) = 1 on [0,
µ), where
µ := sup{y :P{Xi < y}= 0};
a(t) ∈ (0,1) on (√µ,1); and a(t) = 0 on (1,∞).
Furthermore, the recent results of the author [26] allow us to prove the
conjecture from Fact 1 that aPoiss(t) = aUnif(t) = 1− t2 for 0≤ t≤ 1. There
is an amazing contrast between the simplicity of this formula and the hard
calculations one needs to obtain it. It is remarkable that now we know the
limit function a(t) for the main models of initial positions.
Our main goal is to improve (1) by finding the next term in the asymp-
totics of Kn(t). The result is the following statement, where the standard
symbol
D−→ denotes weak convergence and D denotes the Skorohod space.
Theorem 1. In the i.d. model with continuous Xi satisfying EX
for some γ > 4, there exists a centered Gaussian process K(·) on [0,1) such
Kn(·)− na(·)√
D−→K(·) in D[0,1− ε] for all ε ∈ (0,1)(2)
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 5
as n→ ∞. The process K(·) depends on the distribution of Xi. This pro-
cess satisfies K(0) = 0 and has a.s. continuous trajectories. The covariance
function R(s, t) of K(·) is continuous on [0,1)2, R(s, t)> 0 on (√µ,1)2, and
R(s, t) = 0 on [0,1)2 \ (√µ,1)2.
In the uniform model, (2) holds for some centered Gaussian process KUnif(·)
on [0,1). This process satisfies KUnif(0) = 0 and has a.s. continuous trajecto-
ries. The covariance function RUnif(s, t) of KUnif(·) is continuous on [0,1)2,
and RUnif(s, t) =RPoiss(s, t)− s2t2.
Thus, the Poisson and the uniform models lead to different limit processes
KPoiss(·) and KUnif(·), although aPoiss(·) = aUnif(·).
As an immediate corollary of Theorem 1 (see Billingsley [3], Section 15),
we get
Kn(t)− na(t)√
D−→N (0, σ2(t)), n→∞(3)
for any t < 1, where σ2(t) := R(t, t). It is possible to show that in the i.d.
model, (3) holds for all t 6= 1 under the less restrictive condition EX2i <∞,
with σ2(t) = 0 for t > 1; continuity of Xi is not required.
We also study convergence of the left-hand side of (3) at the critical
moment t= 1. Apparently, the limit is not Gaussian, but this complicated
problem is related to a curious, but hardly provable conjecture on integrated
random walks. In view of this non-Gaussianity, it seems impossible to prove
any extended version of Theorem 1 that describes the weak convergence of
trajectories on the whole interval [0,1]; we refer to Section 7 for further
discussion.
We finish this subsection with a note on scaling. In our model, the masses
of particles are equal to 1
and the distances between them are of the order
. Let us rescale the i.d. model by multiplying all masses and distances
by n: the system of particles of mass one each, initially located at points
S1 − S[n/2], S2 − S[n/2], . . . , Sn − S[n/2], is called the expanding model. The
particles are shifted by S[n/2] because we want the system to expand “filling”
the whole line as n→∞ rather than only the positive half-line.
All results of our paper hold true for the expanding model. This is not
unexpected because the shift does not produce any changes and the rescaling
of masses is equivalent to the time contraction by n times while the rescaling
of distances is equivalent to the time expansion by n times. We refer the
reader to Section 2 below or to Lifshits and Shi [16] for rigorous arguments.
1.3. Organization of the paper. In Section 2 we describe a general method
which is used to study systems of sticky particles. This method is applied for
studying the i.d. model in Section 3, where we investigate some properties of
6 V. V. VYSOTSKY
the aggregation process. We will show that the aggregation process is highly
local, that is, the behavior of a particle is essentially defined by the motion
of neighbor particles. This localization property suggests that we could use
limit theorems for weakly dependent variables to prove both Fact 1 and
Theorem 1 for the i.d. model; this will be done in Section 4. Then we will
prove Theorem 1 for the uniform model in Section 5. In Section 6 we study
the number of clusters at the critical moment t = 1. Some open questions
are discussed in Section 7.
2. Method of barycenters. In this section we briefly describe the method
of barycenters, which is the main tool used to study systems of sticky par-
ticles; it is also applicable to more general models where particles could
have nonzero initial speeds and different masses. The method of barycenters
was independently introduced by E, Rykov and Sinai [6] and Martin and
Piasecki [20].
Let us start with several definitions. We always numerate particles from
left to right and identify particles with their numbers. A block of particles is
a nonempty set J ⊂ [1, n] consisting of consecutive numbers. For example,
the block (i, i+k] consists of particles i+1, . . . , i+k. Note that there are not
any relations between blocks and clusters: for example, a block’s particles
could be contained in different clusters and these clusters could even contain
particles that do not belong to the block.
It is convenient to assume that initial particles do not vanish at collisions
but continue to exist in created clusters. Then the coordinate xi,n(t) of a
particle i could be defined as the coordinate of a cluster that contains the
particle at time t. The second subscript n always indicates the number of
initial particles; we will omit this subscript as often as possible.
By xJ(t) := |J |−1
i∈J xi(t) denote the position of the barycenter of a
block J at time t. Further, define
x∗J(t) := xJ(0) +
where M
J := n
−1(n −maxj∈J) and M (L)J := n−1(minj∈J −1) are the to-
tal masses of particles located to the right and to the left of the block J ,
respectively.
A block is free from the right up to time t if, up to this time, the block’s
particles did not collide with particles initially located to the right of the
block. We similarly define blocks that are free from the left and say that a
block is free up to time t if it is both free from the right and from the left.
The next statement plays the key role in the analysis of sticky particles
systems. The barycenter of a free block moves as an imaginary particle con-
sisting of all particles of the block put together at the initial barycenter. In a
more precise and general way, we state the following.
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 7
Proposition 1. If a block J is free from the right (resp. left) up to time
t, then xJ(s)≥ x∗J(s) for s ∈ [0, t] [resp. xJ(s)≤ x∗J(s)]. If a block J is free
up to time t, then xJ(s) = x
J(s) for s ∈ [0, t].
This statement could be found, for example, in Lifshits and Shi [16],
Proposition 4.1. The easy proof is based on the property of conservation of
momentum.
The moment when a particle j sticks with its right-hand side neighbor
j + 1 is called the merging time Tj,n of the particle j. In other words, Tj,n
is the first moment when particles j and j + 1 are contained in a common
cluster; here j ∈ [1, n− 1]. Proposition 4.3 from Lifshits and Shi [16], which
is stated below, gives us a way to calculate Tj,n.
Proposition 2. For every j ∈ [1, n− 1], we have
Tj,n = min
j<k≤n
0≤l<j
{s≥ 0 :x∗(j,k](s) = x∗(l,j](s)}.(4)
Thus, Tj,n is expressed by means of barycenters. Note that since
x∗(j,k](s)− x∗(l,j](s) = x(j,k](0)− x(l,j](0)−
s2,(5)
each of the equations x∗
(j,k]
(s) = x∗
(l,j]
(s) has a unique nonnegative solution.
We also mention that at the moment Tj,n appears a cluster that consists of
the particles l+1, . . . , k, where k and l are minimizers of the right-hand side
of (4).
We will prove Proposition 2 since the proof is simple and perfectly illus-
trates the sense of the method of barycenters.
Proof of Proposition 2. For any u < Tj,n, the particles j and j + 1
are contained in different clusters. Therefore, for every l < j, the block [l, j]
is free from the right up to time u, and for every k > j, the block [j + 1, k]
is free from the left. By Proposition 1,
x∗(l,j](u)≤ x(l,j](u)≤ xj(u)<xj+1(u)
≤ x(j,k](u)≤ x∗(j,k](u),
and since, by (5), the function x∗
(j,k]
(s)− x∗
(l,j]
(s) is decreasing for s≥ 0, we
conclude that
u <{s≥ 0 :x∗(j,k](s) = x∗(l,j](s)}.
Taking minimum over k, l and taking supremum over u, we get Tj,n ≤
min{· · ·}.
8 V. V. VYSOTSKY
Let us prove the last inequality in the other direction. By the definition of
Tj,n, there exist an l < j and a k > j such that the blocks (l, j] and (j, k] are
free up to time Tj,n (clusters containing particles from these blocks collide
exactly at time Tj,n). In view of Proposition 1,
x∗(l,j](Tj,n) = x(l,j](Tj,n) = x(j,k](Tj,n) = x
(j,k](Tj,n);
hence Tj,n = {s≥ 0 :x∗(j,k](s) = x
(l,j]
(s)} and Tj,n ≥min{· · ·}. �
3. Study of the i.d. model. The localization property. At first, note that
Kn(t) = 1+
1{t<Ti,n}(6)
because the total number of clusters decreases by one at each moment Ti,n.
This representation plays the key role in the investigation of Kn(t). Clearly,
we need to study properties of the r.v.’s Ti,n to prove limit theorems for
Kn(t); such study will be done in this section.
3.1. The initial study. Let us simplify the representation for Tj,n from
Proposition 2. In this section we consider the i.d. model of initial positions,
where xj,n(0) =
Sj . Recall that Sj is a random walk with i.i.d. increments
{Xj}j∈Z (we will need the variables {Xj}j≤0 later).
Rewrite the initial distance between barycenters as
x(j,k](0)− x(l,j](0)
i=j+1
j − l
i=l+1
i=j+1
(Si − Sj+1) +
j − l
i=l+1
(Sj − Si) + (Sj+1 − Sj)
k−j−1
(Sj+i+1 − Sj+1) +
j − l
j−l−1
(Sj − Sj−i) +Xj+1
let us agree that
:= 0. Further, by
x(j,k](0)− x(l,j](0)
k−j−1
j+i+1
m=j+2
j − l
j−l−1
m=j−i+1
Xm +Xj+1
k−j−1
(k− j − i)Xj+i+1
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 9
j − l
j−l−1
(j − l− i)Xj−i+1 +Xj+1
and (5), we have
x∗(j,k](s)− x∗(l,j](s) =
Fk−j,j,j−l(s),
where
Fp,j,q(s) :=
(p− i)Xj+i+1
(q − i)Xj−i+1 +Xj+1 −
(for p, q ≥ 1 and j ∈ Z). Now, by Proposition 2, we get
Tj,n = min
j<k≤n
0≤l<j
{s≥ 0 :Fk−j,j,j−l(s) = 0}
= min
1≤k≤n−j
1≤l≤j
{s≥ 0 :Fk,j,l(s) = 0}.(8)
Note that Fp,j,q(0) ≥ 0 for all p, j, q and Fp,j,q(s) is decreasing for s ≥ 0.
This function could be also written in the more convenient form:
Fp,j,q(s) =
(p− i)(Xj+i+1 − s2)
(q − i)(Xj−i+1 − s2) + (Xj+1 − s2).
3.2. Localization property of the aggregation process. We see that Tj,n
is a function of X2, . . . ,Xn; in other words, it is necessary to know the
distances between all n particles to find Tj,n. The aggregation process is
actually highly local, that is, the value of Tj,n is essentially defined by the
initial distances between neighbor particles {i} of j for which |j − i| is small
enough.
To make this statement rigorous, we need to introduce the following no-
tation. Let us put
j := min
1≤k,l≤M
{s≥ 0 :Fk,j,l(s) = 0}, j ∈ Z,M ∈N,
which is expressed in terms of the variables {Xi}|j−i|≤M only. Also, define
Tj := inf
k,l≥1
{s≥ 0 :Fk,j,l(s) = 0}, j ∈ Z,
10 V. V. VYSOTSKY
which is, in some sense, the merging time in an appropriate infinite system
of particles. The reader could construct such system by considering the limit
of the expanding model, see Section 1.
It is clear that
Tj ≤ Tj,n ≤ T (j∧n−j)j , j, n ∈N, j ≤ n,(10)
where by ∧ and ∨ we denote minimum and maximum, respectively, and
Tj ≤ T (M)j , j ∈ Z,M ∈N.(11)
Let us estimate the rate of the convergence of P{Tj 6= T (M)j } to zero as the
“radius of the neighborhood” M tends to infinity. We thus could “measure”
the above-mentioned locality of the aggregation process. In fact, by (10), we
have P{Tj,n 6= T (M)j } ≤ P{Tj 6= T
j } for any n ∈N, j ≤ n, andM ≤ j∧n−j.
Lemma 1. Suppose EX
i <∞ for some γ ≥ 1. Then there exists a non-
decreasing function ρ(t) such that
max(P{1{t≤Tj} 6= 1{t≤T (M)
}},P{Tj 6= T
j , T
j ≤ t})≤ ρ(t)M
1−γ(12)
for any t ∈ (0,1), j ∈ Z, and M ∈N. Moreover, for any t < 1, the left-hand
side of (12) is o(M1−γ).
Proof. Let us estimate the first probability in the left-hand side of
(12). By properties of Fk,j,l(·) and definitions of T (M)j and of Tj ,
P{1{t≤Tj} 6= 1{t≤T (M)
}}= P{Tj < t≤ T
k,l≥1
Fk,j,l(t)< 0, min
1≤k,l≤M
Fk,j,l(t)≥ 0
By (9), this expression does not depend on j, and putting j :=−1,
P{1{t≤Tj} 6= 1{t≤T (M)
(k− i)(Xi − t2)
+ inf
(l− i)(X−i − t2) + (X0 − t2)< 0,
1≤k≤M
(k− i)(Xi − t2)
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 11
+ min
1≤l≤M
(l− i)(X−i − t2) + (X0 − t2)≥ 0
We then compare the inequalities in the braces and obtain
P{1{t≤Tj} 6= 1{t≤T (M)
(k− i)(Xi − t2)< min
1≤k≤M
(k− i)(Xi − t2)
(Si − it2)< min
1≤k≤M
(Si − it2)
(Si − it2)< min
k∈{1,M}
(Si − it2)
Now rewrite the event in the last line as
∃k >M : 1
(Si − it2)<min
(Si − it2)
∃k >M : 1
(Si − it2)
(Si − it2)<min
(Si − it2)
Analyzing both cases 0≤ 1
i=1 (Si − it2) and 0> 1M
i=1 (Si − it2), we
conclude that the considered event implies
∃k >M : 1
(Si − it2)< 0
∃k >M :
(Si − it2)< 0
Clearly, the latter implies
{∃i≥M :Si − it2 < 0}=
hence, combining all the estimates together, we get
P{1{t≤Tj} 6= 1{t≤T (M)
}} ≤ 2P
.(13)
Note that we obtained (13) without any assumptions on the moments of Xi.
We now estimate the right-hand side of (13); recall that EXi = 1. Then
the first part of (12) immediately follows from the classical result of Baum
and Katz [1] (see their Theorem 3 and Lemma):
12 V. V. VYSOTSKY
Fact 2. If EXi = a and E|Xi|γ <∞ for some γ ≥ 1, then
= o(k1−γ), k→∞
for any ε > 0. In addition, the series
k=1P{supi≥k |Sii − a|> ε} converges
for all ε > 0 if γ = 2.
The estimation of the second probability in the left-hand side of (12) is
completely analogous, since
{Tj 6= T (M)j , T
j ≤ t}
= {Tj < T (M)j ≤ t}
1≤k,l
Fk,j,l(T
j )< 0, min
1≤k,l≤M
Fk,j,l(T
j ) = 0, T
j ≤ t
We put j :=−1, repeat the estimates, and get
P{Tj 6= T (M)j , T
j ≤ t} ≤ 2P{∃i≥M :Si − i[T
< 0, T
−1 ≤ t}
instead of (13). The right-hand side does not exceed 2P{∃i≥M : Si − it2 <
0}, hence
P{Tj 6= T (M)j , T
j ≤ t} ≤ 2P
.(14)
3.3. The distribution function of T0 in the Poisson model. It is amazing
that in the Poisson model, the distribution function of T0 could be found
explicitly. This is important because by (27) below, the limit function a(t)
equals P{T0 > t} for the i.d. model. Also, in the proof of Theorem 1 for the
uniform model, we will need aPoiss(t) = P{TPoiss0 ≥ t} to be twice differen-
tiable and have a continuous second derivative.
Lemma 2. In the Poisson model, for 0≤ t≤ 1, we have
P{T0 ≥ t}= 1− t2.(15)
In addition, for t≥ 0, n≥ 2, and 1≤ j ≤ n− 1, we have
P{Tj,n ≥ t}= et
1≤k≤j
(Si − it2)≥ 0
1≤k≤n−j
(Si − it2)≥ 0
where Si is a standard exponential random walk.
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 13
Proof. We start with (16). By (8), (9) and properties of Fk,j,l(·),
P{Tj,n ≥ t}= P
1≤k≤n−j
1≤l≤j
Fk,j,l(t)≥ 0
1≤k≤n−j
(k− i)(Xj+i+1 − t2)(17)
+ min
1≤l≤j
(l− i)(Xj−i+1 − t2) +Xj+1 − t2 ≥ 0
In the right-hand side of the last equality, by Y denote the first minimum
and by Ỹ denote the second one.
Suppose X is a standard exponential r.v., Z is a nonnegative r.v., and
that X and Z are independent; then
P{Z ≤X}=
P{Z ≤ x}e−x dx
E1{Z≤x}e
−x dx= E
1{Z≤x}e
−x dx= Ee−Z .
Hence in view of independence of Y , Ỹ , Xj+1 we get
P{Y + Ỹ +Xj+1 − t2 ≥ 0}= EeY+Ỹ−t
EeY−t
EeỸ−t
and therefore,
P{Tj,n ≥ t}= et
P{Y +Xj+1 − t2 ≥ 0} · P{Ỹ +Xj+1 − t2 ≥ 0}.
Now, by
P{Ỹ +Xj+1 − t2 ≥ 0}
1≤l≤j
(l− i)(Xj−i+1 − t2) +Xj+1 − t2 ≥ 0
1≤l≤j
(l− i)(Xi+1 − t2) + l(X1 − t2)
1≤l≤j
(l− i+1)(Xi − t2)≥ 0
we conclude the proof of (16). Indeed, the expression in the last line equals
the first probability in the right-hand side of (16).
14 V. V. VYSOTSKY
Now let us prove (15). From the definition of T0 and T
0 we see that
1{t≤T (k)
} → 1{t≤T0} a.s. as k→∞; then by (16),
P{T0 ≥ t}= et
(Si − it2)≥ 0
Then we need to check that
(Si − it)≥ 0
1− te−t/2
for 0 ≤ t ≤ 1. The complicated calculations of this probability take more
then ten pages. Therefore, they were separated into independent paper [26].
Although these calculations seem to be technical, they are based on quite
original ideas. �
3.4. Some properties of the variables Ti. In this subsection we prove
several important properties of the r.v.’s Ti.
1. The sequence Ti is stationary.
Proof. This statement immediately follows from the definition of Ti and
stationarity of Xi, which are i.i.d.
2. The common distribution function of Ti is defined by
P{Ti ≥ t}= P
(k− i)(Xi − t2)
+ inf
(l− i)(X−i − t2) + (X0 − t2)≥ 0
Proof. This formula follows from (9).
3. We have P{√µ ≤ Ti ≤ 1} = 1 while sup{y :P{Ti < y} = 0} =
µ and
inf{y :P{Ti < y} = 1} = 1; recall that µ = sup{y :P{Xi < y} = 0}. In addi-
tion, if 0<DXi <∞, then P{Ti = 1}= 0.
Proof. First, P{√µ ≤ Ti} = 1 is trivial, because both infima in (19) are
nonpositive.
Second, fix a t≥ 1 and consider P{Ti ≥ t}. Taking into account that infima
in (19) are nonpositive, we obtain
P{Ti ≥ t} ≤ P
(k− i)(Xi − t2) + (X0 − t2)≥ 0
Then by the same arguments as in (18),
P{Ti ≥ t} ≤ P
(k − i+1)(Xi − t2)≥ 0
(Si − it2)≥ 0
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 15
By the strong law of large numbers, this probability is zero for all t > 1.
If t= 1 and 0<DXi <∞, then
(Si − i)≥ 0
= lim
1≤k≤n
(Si − i)≥ 0
= lim
1≤k≤n
Si − i√
and from the invariance principle, we get
P{Ti ≥ 1} ≤ P
0≤s≤1
W (u)du≥ 0
It follows from the asymptotics of unilateral small deviation probabilities of
an integrated Wiener process, see (43) and (44) below, that the last expres-
sion equals zero.
Third, sup{y :P{Ti < y}= 0}=
µ and inf{y :P{Ti < y}= 1} = 1 follow
if we prove that for any t < EXi = 1, the common distribution of the i.i.d.
infima in (19) has an atom at zero. But we have
(k− i)(Xi − t2) = 0
(Si − it2) = 0
and it could be shown via the strong law of large numbers that the last
probability is strictly positive for all t < 1.
4. Suppose Xi is continuous. Then T
j and Tj,n are continuous for any
j, k,n and the common distribution of Tj could have an atom only at 1. In
addition, if EX2i <∞, then Tj are continuous.
Proof. By (7) and (8),
Tj,n = min
1≤k≤n−j
1≤l≤j
H(k, j, l),(20)
where
H(p, j, q) :=
(p− i)Xj+i+1 +
(q − i)Xj−i+1 +Xj+1
Hence Tj,n is continuous as a minimum of a finite number of continuous
r.v.’s. The T
j are also continuous because T
= Tk,2k.
16 V. V. VYSOTSKY
Now we prove the continuity of Tj . By Property 3, it only remains to
verify that P{Tj ≥ t} is continuous on [0,1). But P{T (k)j ≥ t} − P{Tj ≥ t}=
P{1{t≤Tj} 6= 1{t≤T (k)
}}, and in view of (13),
0≤t≤s
|P{T (k)j ≥ t}−P{Tj ≥ t}| ≤ sup
0≤t≤s
for every s < 1 = EXi. The last expression tends to zero by the strong law
of large numbers; then P{Tj ≥ t} is continuous on [0, s] as a uniform limit
of continuous functions P{T (k)j ≥ t}. Since s < 1 is arbitrary, P{Tj ≥ t} is
continuous on [0,1).
5. The cov(1{s≤T0},1{t≤Tk}) tends to zero as k→∞ for all s, t ∈ [0,1). If,
in addition, EX
i <∞ for some γ > 1, then for any s, t ∈ [0,1) and k ∈ N,
we have
|cov(1{s≤T0},1{t≤Tk})| ≤ 2
γ(ρ(s) + ρ(t))k1−γ .(21)
Proof. The idea is to approximate 1{s≤T0} and 1{t≤Tk} by 1{s≤T (k/2)0 }
1{t≤T (k/2)
}, respectively; here by k/2 we mean ⌈k/2⌉, where ⌈x⌉=min{m ∈
Z :m≥ x}. Note that 1{s≤T (k/2)0 }
and 1{t≤T (k/2)
} are independent because the
first is a function of {Xi}i≤k/2 while the second is a function of {Xi}i≥k/2+1.
We then have
|cov(1{s≤T0},1{t≤Tk})|
= |cov(1{s≤T0},1{t≤Tk})− cov(1{s≤T (k/2)
},1{t≤T (k/2)
≤ |E(1{s≤T0}1{t≤Tk} − 1{s≤T (k/2)
}1{t≤T (k/2)
+ |E(1{s≤T0} − 1{s≤T (k/2)
})|+ |E(1{t≤Tk} − 1{t≤T (k/2)
})|(22)
= P{1{s≤T0}1{t≤Tk} 6= 1{s≤T (k/2)
}1{t≤T (k/2)
+ P{1{s≤T0} 6= 1{s≤T (k/2)0 }}+ P{1{t≤Tk} 6= 1{t≤T (k/2)k }
P{1{s≤T0}1{t≤Tk} 6= 1{s≤T (k/2)
}1{t≤T (k/2)
≤ P{1{s≤T0} 6= 1{s≤T (k/2)
} ∪ 1{t≤Tk} 6= 1{t≤T (k/2)
therefore the result follows from Lemma 1.
6. The r.v.’s {Ti}i∈Z, {T (k)i }i∈Z, and {Ti,n}
i=1 are associated ; the author
owes this observation to M. A. Lifshits.
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 17
Proof. Let us first recall the definition and some basic properties of as-
sociated variables. R.v.’s ξ1, . . . , ξm are associated if for any coordinate-wise
nondecreasing functions f, g :Rm →R, it is true that
cov(f(ξ1, . . . , ξm), g(ξ1, . . . , ξm))≥ 0
(assuming that the left-hand side is well defined). An infinite set of r.v.’s is
associated if any finite subset of its variables is associated.
The following sufficient conditions of association are well known; see [7].
(a) Independent variables are associated.
(b) Coordinate-wise nondecreasing functions (of finite number of argu-
ments) of associated r.v.’s are associated.
(c) If the variables ξ1,k, . . . , ξm,k are associated for every k and (ξ1,k, . . . ,
ξm,k)
D−→ (ξ1, . . . , ξm) as k→∞, then ξ1, . . . , ξm are associated.
(d) If two sets of associated variables are independent, then the union of
these sets is also associated.
Then {Ti,n}n−1i=1 are associated for every n by (a), (b) and (20). Analo-
gously, {T (k)i }i∈Z are associated for every k. Finally, since T
i → Ti a.s. as
k→∞ for every i, (c) ensures the association of {Ti}i∈Z.
7. For any s, t ∈R and k ∈ Z,
cov(1{T0≤s},1{Tk≤t})≥ 0.(23)
Proof. This inequality follows from cov(1{T0≤s},1{Tk≤t}) = cov(1{s<T0},
1{t<Tk}), the association of T0, Tk and (b).
8. If EX
i <∞ for some γ ≥ 2, then the stationary sequence min{Ti, t}
is strongly mixing for any t < 1 and its coefficients of strong mixing α(k)
satisfy α(k) = o(k2−γ).
Proof. Recall that stationary r.v.’s ξi are strongly mixing if α(k)→ 0 as
k→∞, where α(k) are the coefficients of strong mixing defined as
α(k) := sup
A∈F0−∞,B∈F
|P(AB)− P(A)P(B)|;
here F0−∞ := σ(ξ0, ξ−1, . . .) and F∞k := σ(ξk, ξk+1, . . .) are the σ-algebras of
“past” and “future,” respectively. It is readily seen that
α(k)≤ sup
0≤f,g≤1
| cov(f(ξ0, ξ−1, . . .), g(ξk, ξk+1, . . .))|,(24)
where the supremum is taken over Borel functions f, g :R∞ → [0,1].
Let us estimate α(k) in the same way we estimated the left-hand side
of (21). Fix some Borel functions f, g : R∞ → [0,1]. We approximate the
variables from the “past” T0∧ t, T−1∧ t, T−2∧ t, . . . by T (k/2)0 ∧ t, T
(k/2+1)
−1 ∧ t,
(k/2+2)
−2 ∧ t, . . . , respectively; and for the variables from the “future,” we
18 V. V. VYSOTSKY
use the analogous approximation. Now, f(T
(k/2)
0 ∧ t, T
(k/2+1)
−1 ∧ t, . . .) and
(k/2)
k ∧ t, T
(k/2+1)
k+1 ∧ t, . . .) are independent because the first is a function
of {Xi}i≤k/2 and the second is a function of {Xi}i≥k/2+1. We then argue in
the same way as in (22) to get
| cov(f(T0 ∧ t, T−1 ∧ t, . . .), g(Tk ∧ t, Tk+1 ∧ t, . . .))|
(T−i ∧ t) 6= (T (k/2+i)−i ∧ t)
(Tk+i ∧ t) 6= (T (k/2+i)k+i ∧ t)
i=k/2
P{(T0 ∧ t) 6= (T (i)0 ∧ t)}.
Now, by the formula of total probability, we have
P{(T0 ∧ t) 6= (T (i)0 ∧ t)}
= P{(T0 ∧ t) 6= (T (i)0 ∧ t), T
0 ≥ t}+ P{(T0 ∧ t) 6= (T
0 ∧ t), T
0 < t}
≤ P{1{t≤T0} 6= 1{t≤T (i)
}}+ P{T0 6= T
0 , T
0 ≤ t}
and combining all the estimates together, by Lemma 1 (24) and arbitrariness
of f and g, we get α(k)≤ 8
i=k/2 o(i
1−γ) = o(k2−γ) if γ > 2. For γ = 2, we
get α(k)≤ 16
i=k/2 P{inf i≥M Sii < t
2}= o(1) using the same argument and
applying (13), (14), and Fact 2 instead of Lemma 1.
3.5. The last collision. We finish this section with a statement on the
convergence of the moments of the last collision.
Proposition 3. In the i.d. model, T lastn
P−→ 1 as n→∞ if EX2i <∞.
This result is well known for the Poisson model; see Giraud [8].
Proof of Proposition 3. Let us first prove that P{T lastn ≥ t}→ 0 as
n→∞ for all t > 1. Since T lastn =max1≤j≤n−1Tj,n, we have
P{T lastn ≥ t} ≤
P{Tj,n ≥ t}.(25)
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 19
By taking into account that the minima in (17) are nonpositive and by
arguing as in (18),
P{Tj,n ≥ t} ≤ P
1≤k≤j∨n−j
(k− i)(Xj+i+1 − t2) +Xj+1 − t2 ≥ 0
1≤k≤j∨n−j
(k− i+1)(Xi − t2)≥ 0
1≤k≤n/2
(Si − it2)≥ 0
We claim that (without any assumptions on the moments of Xi)
P{Tj,n ≥ t} ≤ P
i≥(t−1)/4tn
1 + t2
;(26)
recall that t > 1. Clearly, (26) follows if we check that
1≤k≤n/2
(Si − it2)≥ 0
i≥(t−1)/4tn
1 + t2
Assume the converse; then, by the nonnegativity of Si,
(Si − it2) =
(Si − it2) +
i=cn+1
(Si − it2)
(Scn − it2) +
i=cn+1
1 + t2
− it2
where c := t−1
. We estimate the last expression with
cnScn −
(cn)2
t2 − (n/2)
2 − (cn)2
2 − 1
n2 − 1/4− c
2 − 1
It is simple to check that the right-hand side is negative, thus we have a
contradiction.
Then from (25), (26) and Fact 2 it follows that P{T lastn ≥ t} =
i=1 o((cn)
−1) = o(1) for all t > 1.
Now let us prove that P{T lastn < t} → 0 as n → ∞ for all t < 1. Since
T lastn =max1≤j≤n−1Tj,n, we estimate
P{T lastn < t} ≤ P
n,n < t
P{1{t≤Tj√n,n} 6= 1{t≤T (
20 V. V. VYSOTSKY
In view of (10) and Lemma 1, the sum is
j=1 o(n
−1/2) = o(1), hence it
remains to check that the first probability in the last line tends to zero.
For a fixed n, all T
are independent because each one is a function of
{Xi}|j√n−i|≤√n/2 (to be precise, of Xj√n−√n/2+2, . . . ,Xj√n+√n/2). Thus,
n−1{T (
n/2)√
< t} ≤ P
n−1{T0 < t},
which tends to zero; indeed, P{T0 < t}< 1 by Property 3, Section 3.4. �
4. Proofs of Fact 1 and Theorem 1 for the i.d. model. Recall that the
number of clusters Kn(t) is given by (6). Our idea is to study
i=1 1{t<Ti}
instead of
i=1 1{t<Ti,n}: We thus deal with a single sequence Ti and avoid
considering the triangular array Ti,n.
Let us now prove Fact 1 for the i.d. model. We prove (1) for t 6= 1 without
any additional assumptions on Xi; for t = 1, we require EX
i < ∞. The
properties of the limit function a(t) were studied in Section 3.4, Properties
3 and 4.
Proof of Fact 1. We put
a(t) := P{T0 > t}.(27)
Let us first prove (1) for all t < 1. It is sufficient to check that
Kn(t)
1{t<Ti}
P−→ 0, n→∞.(28)
Indeed, the stationary sequence 1{t<Ti} satisfies the law of large numbers by
Property 5, Section 3.4, and the well-known result of S. N. Bernstein:
Fact 3. The law of large numbers holds for r.v.’s ξi if there exists a
sequence r(k)→ 0 such that cov(ξi, ξj)≤ r(|i− j|) for all i, j ∈N.
By (6),
Kn(t)
1{t<Ti}
(1{t<Ti,n} − 1{t<Ti}),
where we used (10) to get the nonnegativity of the right-hand side. Then
(28) immediately follows from the Chebyshev inequality provided that the
expectation of the right-hand side tends to zero. By using (10), we obtain
E(1{t<Ti,n} − 1{t<Ti})≤
(E1{t<T (i∧n−i)
} − E1{t<Ti})
P{1{t<Ti} 6= 1{t<T (i∧n−i)
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 21
which is 2
i=1 o(1) = o(1) by Lemma 1. To be very precise, Lemma 1
deals with slightly different indicators, but we can estimate the considered
probability by repeating the proof of Lemma 1 word for word (or just use
Property 4, Section 3.4).
We now check that (1) holds for all t > 1. Using (26) gives E
Kn(t)
i=1 P{Ti,n > t} → 0 as n→∞ and
Kn(t)
P−→ a(t) = 0 follows from the
Chebyshev inequality.
It remains to check that (1) holds for t = 1 if EX2i < ∞ to conclude
the proof. If DXi = 0, then the situation is deterministic, this case was
described in Introduction. Here we always have Kn(1) = 1 and (1) is true.
If 0<DXi <∞, then by Property 3 from Section 3.4, we have a(1) = 0 and
P{T0 = 1} = 0; consequently, a(t) = P{T0 > t} is continuous at t= 1. Then
(1) is true for t = 1 since 0 <
Kn(1)
≤ Kn(t)
P−→ a(t) for any t ∈ (0,1) and
a(t)→ a(1) = 0 as tր 1. �
Now we prove Theorem 1 for the i.d. model. We think of D[0,1] as of a
separable metric space equipped with the Skorohod metric d, which induces
the Skorohod topology.
Proof of Theorem 1. At first, we prove (2). In view of representation
(6) for Kn(t), relation (2) follows from the relation
0≤t≤1−ε
1{t<Ti,n} −
1{t<Ti}
P−→ 0 for all ε ∈ (0,1)(29)
and the existence of a centered Gaussian process K(·) on [0,1) such that
1{t<Ti} −na(t)
D−→K(·) in D[0,1− ε] for all ε ∈ (0,1).(30)
Indeed, if Yn
D−→ Y and d(Yn, Y ′n)
P−→ 0 for some random elements Yn, Y ′n, Y
of the separable metric spaceD[0,1−ε], then Y ′n
D−→ Y ; recall that d(Yn, Y ′n)≤
supt∈[0,1−ε] |Yn(t)− Y ′n(t)|.
We start with (29). It is sufficient to prove that the expectation of the
left-hand side tends to zero. Since the supremum of a sum does not exceed
the sum of suprema, let us check that
E sup
0≤t≤1−ε
|1{t<Ti,n} − 1{t<Ti}| −→ 0 for all ε ∈ (0,1).(31)
By (10), we have
E sup
0≤t≤1−ε
|1{t<Ti,n} − 1{t<Ti}| ≤ E sup
0≤t≤1−ε
(1{t<T (i∧n−i)
} − 1{t<Ti})
22 V. V. VYSOTSKY
= P{Ti 6= T (i∧n−i)i , Ti ≤ 1− ε}
= P{Ti 6= T (i∧n−i)i , T
(i∧n−i)
i < 1− ε}
+ P{1{1−ε≤Ti} 6= 1{1−ε≤T (i∧n−i)
where the last equality was obtained via the formula of total probability.
Combining the estimates together and using Lemma 1,
E sup
0≤t≤1−ε
|1{t<Ti,n} − 1{t<Ti}|
≤ 2ρ(1− ε)√
(i ∧ n− i)1−γ = 4ρ(1− ε)√
i1−γ .
The last expression is O(n3/2−γ) and (31), which implies (29), follows.
Now let us prove (30). As long as
Un(t) :=−
1{t<Ti} − na(t)
1{Ti≤t} − (1− a(t))
the Un(·) is the empirical process of stationary r.v.’s Ti with the continuous
common distribution function 1− a(t). By K(·) D=−K(·), (30) is equivalent
to the existence of a centered Gaussian process K(·) on [0,1) such that
Un(·)
D−→K(·) in D[0,1− ε] for all ε ∈ (0,1).(32)
We will use the following result from Lin and Lu [17], Section 12 on
convergence of empirical processes. They attribute this statement to Q.-M.
Shao, who published it in 1986, in Chinese.
Fact 4. Let ξi be a sequence of stationary strongly mixing r.v.’s dis-
tributed on [0,1], and let F be the common distribution function of ξi.
Suppose F (x) = x on [0,1] (i.e., ξi are uniformly distributed) and the co-
efficients of strong mixing of the sequence F (ξi) decrease as O(k
−(2+δ))
as k → ∞ for some δ > 0. Then the empirical processes of ξi weakly con-
verge in D[0,1] to a centered Gaussian process with the covariance function
i∈Z cov(1{ξ0≤s},1{ξi≤t}).
Remark. The limit Gaussian process is a.s. continuous on [0,1]. Fact 4
also holds true if F is an arbitrary continuous distribution function.
The a.s. continuity of the limit process could be concluded by a compar-
ison of the proof from Lin and Lu [17] with the proof of Theorem 22.1 from
Billingsley [3]. The statements and the proofs of these theorems are identical,
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 23
but Lin and Lu do not state the continuity while Billingsley does. Further,
since F (ξi) is uniformly distributed on [0,1] if F is continuous, Fact 4 holds
true for every continuous F ; see the proof of Theorem 22.1 by Billingsley [3]
for explanations.
Recall that we need to prove the convergence of the empirical process of
Ti. It seems that the r.v.’s Ti are not strongly mixing; but min{Ti,1− ε} are
strongly mixing because of Property 8, Section 3.4. These variables are not
continuous and so we need to fix them. Let us fix an ε ∈ (0,1), and let αi
be i.i.d. r.v.’s independent of all Ti and, say, uniformly distributed on [0, ε];
we define T̃i := min{Ti,1− ε}+ 1{Ti≥1−ε}αi.
The stationary variables T̃i are distributed on [0,1], their common dis-
tribution function G is continuous, and the coefficients of strong mixing
of G(T̃i) decrease as o(k
2−γ). The proof of the last statement is the same
as the proof of Property 8 from Section 3.4. Indeed, approximate the vari-
ables G(T̃0),G(T̃−1), . . . from the “past” by G(T̃
(k/2)
0 ),G(T̃
(k/2+1)
−1 ), . . . where
i := min{T
i ,1− ε}+ 1{T (m)
≥1−ε}αi; use the analogous approximation
for the variables from the “future”; and then repeat word for word the ar-
guments of the previous proof.
Now, recalling that γ > 4, we see that T̃i satisfy the assumptions of Fact 4,
with the only difference that their distribution is not uniform. By Ũn(·)
denote the empirical process of T̃i; clearly, Ũn(·) coincides with the empirical
process Un(·) of Ti on [0,1− ε]. By the remark to Fact 4, we conclude that
first,
Ũn(·)
D−→ K̃(·) in D[0,1],(33)
where K̃(·) is a centered Gaussian process with the covariance function
R̃(s, t) :=
cov(1{T̃0≤s},1{T̃i≤t})
and, second, trajectories of K̃(·) are a.s. continuous on [0,1].
[There exists a simpler and more elegant proof of (33). Note that {T̃i}i∈Z
are associated as coordinate-wise nondecreasing functions of associated r.v.’s
{Ti, αi}i∈Z, see (a), (b) and (d) from Property 6, Section 3.4. Then we can
obtain (33) applying the result of Louhichi [18] on convergence of empirical
processes of stationary associated r.v.’s ξi instead of using Fact 4. This the-
orem requires only cov(F (ξ0), F (ξk)) = O(k
−(4+δ)), which could be proved
analogously to Property 5, Section 3.4. Thus we avoid the complicated esti-
mations of the strong mixing coefficients, and the proof of (33) is becomes
much simpler. The only problem is that this proof requires γ > 5.
We also note that the a.s. continuity of K̃(·) could be proved directly,
without referring to the proof of Fact 4. The arguments should be the same
as in the proof of the continuity of KUnif(·) in Section 5.]
24 V. V. VYSOTSKY
Define
R(s, t) :=
cov(1{T0≤s},1{Ti≤t}),(34)
which is, evidently, equal to R̃(s, t) on [0,1 − ε]2. Since R̃(s, t) is positive
definite and ε > 0 is arbitrary, the function R(s, t) is positive definite on
[0,1)2. Therefore, by Lifshits [15], Section 4, there exists a centered Gaussian
process K(·) on [0,1) with the covariance function R(s, t). The trajectories
of K(·) are a.s. continuous on [0,1) by K(·) D= K̃(·) on [0,1−ε], arbitrariness
of ε > 0, and the a.s. continuity of K̃(·) on [0,1].
Finally, by (33), Ũn(·) = Un(·) on [0,1− ε], K̃(·) D=K(·) on [0,1− ε], and
the a.s. continuity of K̃(·), we get (32). Since (32) implies (30), we conclude
the proof of (2).
Only the stated properties of R(s, t) remain to be proven. The continuity
of the joint distribution function of continuous variables T0 and Ti implies
that cov(1{T0≤s},1{Ti≤t}) is continuous on [0,1)
2 for every i ≥ 0. Then, in
view of (21), R(s, t) is continuous on [0,1)2 as a sum of uniformly converging
series of continuous functions.
The strict positivity of R(s, t) on (
µ,1)2 trivially follows from (34), (23)
and cov(1{T0≤s},1{T0≤t}) = a(s∨ t)(1−a(s∧ t)) > 0; the last inequality holds
by Property 3, Section 3.4. The R(s, t) = 0 on [0,1)2 \ (√µ,1)2 follows from
P{Ti ≤
µ}= 0, which holds by Properties 3 and 4 from Section 3.4. �
We note that (3) holds for t 6= 1 under the less restrictive condition EX2i <
∞. For t < 1, the proof is almost the same: By (29), which is true for γ > 3/2,
we conclude that (3) holds if the stationary associated sequence 1{t<Ti}
satisfies the central limit theorem. Then we refer to the central limit theorem
for stationary associated sequences from Newman [21]; his theorem requires
only R(t, t)<∞, that is, the convergence of the right-hand side of (34). This
condition holds by (13) and Fact 2. For t > 1, relation (3) holds true with
σ2(t) = 0 because of Proposition 3.
Finally, note that the process K(·) is associated, that is, the r.v.’s
{K(t)}t∈[0,1) are associated. In fact, by (6), Property 6 from Section 3.4,
and Condition (b) from the same Property 6, the processes
Kn(·)−na(·)√
associated for every n. Then K(·) is associated by (2) and (c), Property 6.
5. Proof of Theorem 1 for the uniform model. There exists a simple
method that allows to extend results from the Poisson model to the uniform
model and vise versa. The method is based on the next statement (see
Karlin [13], Section 9.1).
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 25
Fact 5. Let Si be an exponential random walk. Then for any k ≥ 1, we
, . . . ,
= (U1,k,U2,k, . . . ,Uk,k),(35)
where Ui,k are the order statistics of k i.i.d. r.v.’s uniformly distributed on
[0,1]. Moreover, the random vector in the left-hand side of (35) is indepen-
dent of Sk+1.
Therefore, if xPoissj,n (0) =
Sj are the initial positions of particles in the
Poisson model, then for the initial positions of particles in the uniformmodel,
we have xUnifj,n (0) =
· xPoissj,n (0). By Proposition 2 and (5), we conclude
TUnifj,n = β
Poiss
j,n , βn :=
,(36)
and hence, using (6), we get
KUnifn (t) =K
Poiss
n (βnt).(37)
Note that the process KUnifn (·) and the r.v. βn are independent since val-
ues of the process are defined by xUnif1,n (0), . . . , x
n,n (0), which are mutually
independent of βn by Fact 5.
Now we prove Theorem 1 for the uniform model.
Proof of Theorem 1. Denote
Yn(t) :=
KUnifn (t)− na(t)√
, Zn(t) :=
n(a(t)− a(βnt));
we stress that Yn(·) and Zn(·) are independent.
Fix an ε ∈ (0,1). First, it follows from (2) for the Poisson model and (37)
Yn(·) +Zn(·)
D−→KPoiss(·) in D[0,1− ε].(38)
Indeed, the process Yn(·) +Zn(·) is obtained from 1√n(K
Poiss
n (·)− na(·)) by
the random time change t 7→ βnt; and since ‖βnt− t‖C[0,1−ε]
P−→ 0, we have
Yn(·) +Zn(·),
KPoissn (·)− na(·)√
P−→ 0
by the definition of the Skorohod metric d.
Second, from Fact 1, (15), and (27) it follows that aUnif(t) = aPoiss(t) =
P{TPoiss0 ≥ t}= 1− t2 for 0≤ t≤ 1, and by the central limit theorem,
Zn(t)
D−→ t2η in D[0,1− ε],(39)
26 V. V. VYSOTSKY
where η is a standard Gaussian r.v.
We claim that (38), the independence of Yn(·) and Zn(·), and (39) yield
the weak convergence of Yn(·) in D[0,1− ε]. Let us check the tightness of
Yn(·) and the convergence of their finite-dimensional distributions.
The tightness of Yn(·) in D[0,1− ε] follows from Yn(·) = (Yn(·)+Zn(·))−
Zn(·), (38), and (39). Indeed, by the Prokhorov theorem, (38) and (39) yield
that both sequences Yn(·) + Zn(·) and −Zn(·) are tight. But trajectories
of −Zn(·) are a.s. continuous because of the continuity of a(·), and the
tightness follows from the continuity of addition + :D×C →D and the fact
that under any continuous mapping, the image of a compact set is also a
compact set.
Now we study convergence of finite dimensional distributions of Yn(·).
Recall that the characteristic function of a centered Gaussian vector in Rm
is e−1/2(Ru,u), where u ∈ Rm and R is the covariance matrix of the vector.
Then (38), the independence of Yn(·) and Zn(·), and (39) yield that for the
characteristic functions of all finite-dimensional distributions of Yn(·), we
Eei(Yn(t),u) −→ e−1/2({R
Poiss(tj ,tk)−t2j t
j,k=1
,(40)
where u ∈Rm, t= (t1, . . . , tm) ∈ [0,1−ε]m, and Yn(t) := (Yn(t1), . . . , Yn(tm)).
We stress that (40) is true for every t ∈ [0,1− ε]m since the limit processes
in (38) and (39) have continuous trajectories.
We see that the matrix {RPoiss(tj , tk)− t2j t2k}mj,k=1 is positive definite for
any t = (t1, . . . , tm) ∈ [0,1− ε]m and m≥ 1 since the absolute value of the
left-hand side of (40) does not exceed one. Putting
RUnif(s, t) :=RPoiss(s, t)− s2t2,
we have {RPoiss(tj , tk)− t2j t2k}mj,k=1 = {RUnif(tj , tk)}mj,k=1; then the function
RUnif(s, t) is positive definite on [0,1)2 since ε > 0 is arbitrary. Thus, by
Lifshits [15], Section 4, RUnif(s, t) is the covariance function of some centered
Gaussian process KUnif(·) on [0,1).
Relation (2) is thus proved. Now check that KUnif(·) ∈C[0,1− ε] a.s. to
conclude the proof of Theorem 1 for the uniform model.
For this purpose, let us prove that a.s., trajectories of Yn(·) have jumps
of size 1√
only. In fact, the jumps of Yn(·) coincide with the jumps of
KUnifn (·), whose jumps are of size 1√n if and only if T
6= TUnifj2,n for
1 ≤ j1 6= j2 ≤ n − 1. By (36), we need to verify that TPoissj1,n 6= T
Poiss
for 1 ≤ j1 6= j2 ≤ n − 1. This relation follows from (20) if H(k1, j1, l1) 6=
H(k2, j2, l2) a.s. for j1 6= j2 and k1, k2, l1, l2 ≥ 1. The last a.s. nonequality is
obvious because if the equality holds true, then a certain nontrivial linear
combination of i.i.d. exponential Xi equals zero.
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 27
Then there exist a.s. continuous Ỹn(·) such that supt∈[0,1−ε] |Ỹn(t)−Yn(t)| ≤
a.s.; consequently, d(Ỹn, Yn) ≤ 1√n a.s. Then by Yn(·)
D−→ KUnif(·), we
also have Ỹn(·)
D−→KUnif(·). But 1 = lim inf P{Ỹn(·) ∈C} ≤ P{KUnif(·) ∈C}
since C ⊂D is closed in the Skorohod topology, therefore, a.s., KUnif(·) is
continuous on [0,1− ε].
Since ε ∈ (0,1) is arbitrary, a.s., KUnif(·) is continuous on the whole inter-
val [0,1). The RUnif(s, t) =RPoiss(s, t)− s2t2 is continuous on [0,1)2 because
RPoiss(s, t) is. �
6. The number of clusters at the critical moment. Now we turn our
attention to the number of clusters at the critical moment t = 1. We are
interested in the behavior of
Kn(1)− na(1)√
Kn(1)√
which is the left-hand side of (3) at t = 1; here we have a(1) = 0 under
EX2i <∞, see Property 3, Section 3.4.
We do not know if this sequence is weakly convergent, but we hope that
it is. We also have a naive guess that its limit is Gaussian because the limit
in Theorem 1 is Gaussian. In view of Kn(1)≥ 1, this conjectured weak limit
is nonnegative, hence it is Gaussian if and only if it is identically equal to
zero. However, the results of this section show that the limit is nonzero, thus
our guess on Gaussianity fails.
The study of convergence of
Kn(1)√
is quite complicated. Therefore, in this
section, we consider only the Poisson model. First, let us prove the following
statement.
Proposition 4. In the Poisson model, we have limn→∞P{Kn(1) =
1}> 0.
Proof. On the one hand, Kn(1) = 1 is equivalent to T
n;Poiss ≤ 1, where
T lastn;Poiss denotes the moment of the last collision in the Poisson model. On
the other hand, a result by Giraud [8] states that in the uniform model,
n(T lastn;Unif − 1)
D−→ sup
0≤x≤1
W (y)dy −
W (y)dy
=: τ,
where
W (·) is a Brownian bridge. Now, by (36), we have T lastn;Unif = β−1n T lastn;Poiss,
hence
n(β−1n T
n;Poiss − 1)
D−→ τ.(41)
28 V. V. VYSOTSKY
But from the central limit theorem and the law of large numbers,
n(β−1n − 1) =−
Sn+1 − n√
Sn+1(
Sn+1 +
D−→ η
,(42)
where η is a standard Gaussian r.v. and Si is a standard exponential ran-
dom walk that defines initial positions of particles. Since, in view of Fact 5,
T lastn;Unif = β
n;Poiss and βn are independent, from (41), (42), and the law
of large numbers it follows that
n(T lastn;Poiss − 1)
D−→ τ − η
= τ +
where τ and η are independent. Thus,
P{Kn(1) = 1}= lim
P{T lastn;Poiss ≤ 1}= P
The main advantage of the Poisson model is that, by Lemma 2 and Prop-
erty 4, Section 3.4 we have P{Tj,n > 1}= epjpn−j , where
pk := P
1≤m≤k
(Si − ESi)≥ 0
and Si is a standard exponential random walk. We say that the sequence of
r.v.’s
i=1(Si−ESi) is an integrated random walk. In the proof of Property
3, Section 3.4, we showed that pk → 0 as k→∞. Therefore, it is reasonable
to say that pk are the unilateral small deviation probabilities of an integrated
centered random walk.
We need to obtain the asymptotics of pk → 0 to continue the study of
convergence of
Kn(1)√
. Unfortunately, the results of the rest of this section
are completely dependent on the correctness of the following conjecture.
Conjecture 1. We have pk ∼ c1k−1/4 as k→∞ for some c1 ∈ (0,∞).
Simulations show that the conjecture is true and c1 ≈ 0.36. The weaker
form pk ≍ k−1/4 of Conjecture 1 was proved by Sinai [22], but only for inte-
grated symmetric Bernoulli random walks. It also interesting to note that,
by McKean [19], the unilateral small deviation probabilities of an integrated
Wiener process have the same order as T →∞:
0≤s≤T
W (u)du≥−1
∼ c2T−1/4(43)
for some c2 ∈ (0,∞). The left-hand side of (43) is a unilateral small deviation
probability since
0≤s≤T
W (u)du≥−1
0≤s≤1
W (u)du≥−T−3/2
.(44)
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 29
To be precise, McKean was interested in a more general problem, and
some calculations are required to obtain (43) from his results. Therefore,
we additionally refer to Isozaki and Watanabe [12] who state (43) explic-
itly.
By the results mentioned above, we also suppose that Conjecture 1 is
true for other integrated centered random walks that satisfy some moment
conditions.
Now we are able to prove the following result on convergence of
Kn(1)√
Proposition 5. Suppose Conjecture 1 holds true. Then in the Poisson
model, we have
Kn(1)√
= c3, sup
Kn(1)√
<∞(45)
for some c3 ∈ (0,∞); the sequence Kn(1)√n is tight and uniformly integrable;
and the limit of any weakly converging subsequence of
Kn(1)√
takes value zero
with positive probability, but is not identically equal to zero.
Numerical simulations show that
Kn(1)√
is weakly convergent and that this
convergence is quite fast. In Figure 1 we present the (empirical) distribution
function of
Kn(1)√
for n = 10,000. Since the simulations performed for n =
40,000 showed a hardly perceptible difference, this function seems to be a
good candidate for the distribution function of the conjectured limit.
Note that if we weaken Conjecture 1 to pk ≍ k−1/4, then Proposition 5
still holds true with the only difference that E
Kn(1)√
Proof of Proposition 5. We start with the convergence of the ex-
pectation. On the one hand, by (6) and Lemma 2,
Kn(1)√
pipn−i,
and on the other hand,
i−1/4(n− i)−1/4 = 1
)−1/4(
)−1/4
−→B(3/4,3/4)
as the integral sum of Beta function. Then it follows from Conjecture 1 and
standard arguments that E
Kn(1)√
converges to c3 := ec
1B(3/4,3/4) > 0.
30 V. V. VYSOTSKY
Fig. 1. The distribution function of
Kn(1)√
for n= 10,000.
Now we check the uniform boundedness of E(
Kn(1)√
)2. By (6) it is sufficient
to prove that
i,j=1,i 6=j
P{Ti,n > 1, Tj,n > 1}<∞.(46)
Suppose i < j; then by using (8) and properties of Fk,j,l(·), we get
P{Ti,n > 1, Tj,n > 1}= P
1≤k≤n−i
1≤l≤i
Fk,i,l(1)> 0, min
1≤k≤n−j
1≤l≤j
Fk,j,l(1)> 0
1≤k≤(j−i)/2
1≤l≤i
Fk,i,l(1)> 0, min
1≤k≤n−j
1≤l≤(j−i)/2
Fk,j,l(1)> 0
where by (j − i)/2 we mean ⌈(j − i)/2⌉. The minima in the last expres-
sion are independent as functions of {Xm}m≤(i+j)/2 and {Xm}m≥(i+j)/2+1,
respectively; hence
P{Ti,n > 1, Tj,n > 1} ≤ P
1≤k≤(j−i)/2
1≤l≤i
Fk,i,l(1)> 0
1≤k≤n−j
1≤l≤(j−i)/2
Fk,j,l(1)> 0
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 31
= P{Ti,i+(j−i)/2 > 1} · P{T(j−i)/2,n−j+(j−i)/2 > 1}
= e2pip
⌈(j−i)/2⌉pn−j,
where the first equality follows from (8) and the second follows from Lemma 2.
Recalling Conjecture 1, we get
i,j=1,i 6=j
P{Ti,n > 1, Tj,n > 1} ≤
i,j=1,i 6=j
e2pip
⌈|j−i|/2⌉pn−j
i,j=1,i 6=j
i−1/4⌈|j − i|/2⌉−1/2(n− j)−1/4
i,j=1,i 6=j
)−1/4∣
−1/2(
)−1/4
for some c > 0. The last expression is an integral sum converging to
x−1/4|x− y|−1/2(1− y)−1/4 dxdy,
and it is a simple exercise to check that the integral is finite. This concludes
(46).
The uniform integrability of
Kn(1)√
follows from the second relation from
(45), see Billingsley [3], Section 5, and the tightness follows from the uniform
integrability.
Finally, suppose
Kni(1)√
D−→ ξ for some subsequence ni → ∞ and some
r.v. ξ. Then Eξ = c3 > 0 by the uniform integrability and (45), and hence
ξ is not identically equal to zero. But the distribution of ξ has an atom at
zero since by Proposition 4 and properties of weak convergence,
P{ξ = 0}= lim
P{ξ ≤ ε}
≥ lim
lim sup
Kni(1)√
≥ lim
P{Kni(1) = 1}> 0. �
7. Open questions. 1. The number of clusters at the critical moment
t= 1.
Here the main question is if Conjecture 1 holds true. Even by itself, this
problem is worth studying.
But even if Conjecture 1 is true, we still do not have a proof of weak
convergence of
Kn(1)√
, it is only known that this sequence is tight. The author
32 V. V. VYSOTSKY
strongly believes, relying on numerical simulations, that the limit exists. It
would be interesting to find this conjectured limit, which should be nontrivial
by Proposition 5, in an explicit form.
2. The weak convergence of
Kn(·)−na(·)√
on the whole interval [0,1].
It is very natural to ask if it is possible to strengthen Theorem 1 by
proving the weak convergence of
Kn(·)−na(·)√
in D[0,1]. This complicated
problem returns us again to Question 1 because the weak convergence of
Kn(·)−na(·)√
in D[0,1] implies the weak convergence of
Kn(1)−na(1)√
Kn(1)√
, see
Billingsley [3], Section 15. But even if
Kn(1)√
converges, its weak limit K(1)
is not Gaussian, hence the limit process K(·), which is Gaussian on [0,1),
is no more Gaussian on [0,1]. Therefore, it is doubtful that Theorem 1 is
true in D[0,1]; at least, one should provide a proof completely different from
the presented one. Also, it is unclear how to define the finite-dimensional
distributions of the non-Gaussian K(·) on [0,1] because simulations show
that K(1) would not be independent with K(t) for t < 1.
3. The number of clusters in the warm gas.
In the presented case, initial speeds of particles are zero. This model is of-
ten called the cold gas according to its zero initial temperature. We introduce
a new model stating that initial speeds of particles are anv1, anv2, . . . , anvn,
where vi are some i.i.d. r.v.’s and an is a sequence of normalization con-
stants. This model, called the warm gas, was studied in many papers, for
example, [14, 16, 20, 25].
It is of a great interest to study the behavior of Kn(t) in the warm gas.
In [25], the author proved that in the basic case where an = 1 for all n and
Ev2i <∞, we have
Kn(t)
P−→ 0 for all t > 0. The question is to find a normal-
ization of Kn(t) leading to some nontrivial limit. Clearly, this normalization
depends on an, but it is very possible that there is an effect of phase tran-
sition similar to the one discovered by Lifshits and Shi [16]: If an are small
enough, then the gas has a low temperature and the normalization is the
same as in the cold gas. If an are big enough, as in the basic case an ≡ 1,
then the normalization and the behavior of the gas differ entirely from the
case of the cold gas.
The author believes that the localization property, which is described in
Section 3, could be helpful in a study of these questions.
It is also interesting to compare the behavior of Kn(1) in the warm and in
the cold gases; in the warm gas, the moment t= 1 plays the same “critical”
role as in the cold gas, see Lifshits and Shi [16]. The variable Kn(1) was
studied by Suidan [24], who considered the warm gas with an ≡ 1 and deter-
ministic initial positions of particles (his initial positions were 1
, . . . , n
For this case, Suidan found the distribution of Kn(1) and showed that
EKn(1)∼ logn. Recall that in the presented case, EKn(1)∼ c3
CLUSTERING IN A STOCHASTIC MODEL OF ONE-DIMENSIONAL GAS 33
4. The number of clusters in ballistic systems of sticky particles.
A sticky particles model is called ballistic if it evolves according to the
laws introduced in Section 1, but in the absence of gravitation. Such models
are, in some sense, more natural than gravitational ones because the ba-
sic assumption that gravitation does not depend on distance is sometimes
confusing. However, an unpublished paper of Lifshits and Kuoza shows that
certain gravitational and ballistic models are tightly connected.
It seems interesting to study the number of clusters in the ballistic model.
The author does not know any results in this field.
Acknowledgments. I am grateful to my adviser Mikhail A. Lifshits for
drawing my attention into the subject and for his guidance. I also thank the
anonymous referees for carefully reading this paper and useful comments.
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Department of Probability Theory
and Mathematical Statistics
Faculty of Mathematics and Mechanics
St. Petersburg State University
Bibliotechnaya pl. 2
Stary Peterhof 198504
Russia
E-mail: vlad.vysotsky@gmail.com
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	Introduction
	Description of the model
	Statement of the problem and the results
	Organization of the paper
	Method of barycenters
	Study of the i.d. model. The localization property
	The initial study
	Localization property of the aggregation process
	The distribution function of T0 in the Poisson model
	Some properties of the variables Ti
	The last collision
	Proofs of Fact 1 and Theorem 1 for the i.d. model
	Proof of Theorem 1 for the uniform model
	The number of clusters at the critical moment
	Open questions
	Acknowledgments
	References
	Author's addresses
