bluuebunny/crossref_metadata_2025 huggingface dataset:
A Central Limit Theorem for Contractive Stochastic Dynamical Systems
Authors: Benda Martin
Published: July 2016

If (F

n
)
n∈ℕ is a sequence of independent and identically distributed random mappings from a second countable locally compact state space 𝕏 to 𝕏 which itself is independent of the 𝕏-valued initial variable X
0, the discrete-time stochastic process (X

n
)
n≥0, defined by the recursion equation X

n
= F

n
(X

n−1) for n∈ℕ, has the Markov property. Since 𝕏 is Polish in particular, a complete metric d exists. The random mappings (F

n
)
n∈ℕ are assumed to satisfy ℙ-a.s.

Conditions on the distribution of l(F

n
) are given for the existence of an invariant distribution of X
0 making the process (X

n
)
n≥0 stationary and ergodic. Our main result corrects a central limit theorem by Łoskot and Rudnicki (1995) and removes an error in its proof. Instead of trying to compare the sequence φ (X

n
)
n≥0 for some φ : 𝕏 → ℝ with a triangular scheme of independent random variables our proof is based on an approximation by a martingale difference scheme.
Non-Gaussian Density Processes Arising from Non-Poisson Systems of Independent Brownian Motions
Authors: Feldman Raisa E. , Iyer Srikanth K.
Published: July 2016

The Brownian density process is a Gaussian distribution-valued process. It can be defined either as a limit of a functional over a Poisson system of independent Brownian particles or as a solution of a stochastic partial differential equation with respect to Gaussian martingale measure. We show that, with an appropriate change in the initial distribution of the infinite particle system, the limiting density process is non-Gaussian and it solves a stochastic partial differential equation where the initial measure and the driving measure are non-Gaussian, possibly having infinite second moment.
Exponential functionals of Brownian motion and disordered systems
Authors: Comtet Alain , Monthus Cécile , Yor Marc
Published: July 2016

The paper deals with exponential functionals of the linear Brownian motion which arise in different contexts, such as continuous time finance models and one-dimensional disordered models. We study some properties of these exponential functionals in relation with the problem of a particle coupled to a heat bath in a Wiener potential. Explicit expressions for the distribution of the free energy are presented.
On a formula of Takács for Brownian motion with drift
Authors: Doney R. A. , Yor M.
Published: July 2016

A recent result of Takács (1995) gives explicitly the density of the time spent beforetabove a levelx≠ 0 by Brownian motion with drift. Takács' proof is by means of random walk approximations to Brownian motion, but in this paper we give two different proofs of this result by considerations involving only Brownian motion. We also give a reformulation of Takács' result which involves Brownian meanders, and an extension of Denisov's representation of Brownian motion in terms of two independent Brownian meanders.
Stochastic approximation with non-additive measurement noise
Authors: Chen Han-Fu
Published: July 2016

The Robbins–Monro algorithm with randomly varying truncations for measurements with non-additive noise is considered. Assuming that the function under observation is locally Lipschitz-continuous in its first argument and that the noise is a φ-mixing process, strong consistency of the estimate is shown. Neither growth rate restriction on the function, nor the decreasing rate of the mixing coefficients are required.
The generalization of the Geske–formula for compound options to stochastic interest rates is not trivial–a note
Authors: Frey Rüdiger , Sommer Daniel
Published: July 2016

This note refers to the paper by Geman, El-Karoui and Rochet (1995), in which an extension of the Geske-formula for compound options to the case of stochastic interest rates is proposed. We show that such an extension is not possible in general. However, we point out modifications of Geske's original problem in which closed formulas can still be obtained under stochastic interest rates. In particular we consider the case of an option on a future-style option. Moreover, we sketch a numerical solution to Geske's original problem when interest rates are random.
Sensitivity analysis for Markov reward structures until entrance times
Authors: van Dijk N. M. , Korezlioglu H.
Published: July 2016

This work presents an estimate of the error on a cumulative reward function until the entrance time of a continuous-time Markov chain into a set, when the infinitesimal generator of this chain is perturbed. The derivation of an error bound constitutes the first part of the paper while the second part deals with an application where the time until saturation is considered for a circuit switched network which starts from an empty state and which is also subject to possible failures.
Approximate entropy for testing randomness
Authors: Rukhin Andrew L.
Published: July 2016

This paper arose from interest in assessing the quality of random number generators. The problem of testing randomness of a string of binary bits produced by such a generator gained importance with the wide use of public key cryptography and the need for secure encryption algorithms. All such algorithms are based on a generator of (pseudo) random numbers; the testing of such generators for randomness became crucial for the communications industry where digital signatures and key management are vital for information processing.

The concept of approximate entropy has been introduced in a series of papers by S. Pincus and co-authors. The corresponding statistic is designed to measure the degree of randomness of observed sequences. It is based on incremental contrasts of empirical entropies based on the frequencies of different patterns in the sequence. Sequences with large approximate entropy must have substantial fluctuation or irregularity. Alternatively, small values of this characteristic imply strong regularity, or lack of randomness, in a sequence. Pincus and Kalman (1997) evaluated approximate entropies for binary and decimal expansions of e, π, √2 and √3 with the surprising conclusion that the expansion of √3 demonstrated much less irregularity than that of π. Tractable small sample distributions are hardly available, and testing randomness is based, as a rule, on fairly long strings. Therefore, to have rigorous statistical tests of randomness based on this approximate entropy statistic, one needs the limiting distribution of this characteristic under the randomness assumption. Until now this distribution remained unknown and was thought to be difficult to obtain. To derive the limiting distribution of approximate entropy we modify its definition. It is shown that the approximate entropy as well as its modified version converges in distribution to a χ2-random variable. The P-values of approximate entropy test statistics for binary expansions of e, π and √3 are plotted. Although some of these values for √3 digits are small, they do not provide enough statistical significance against the randomness hypothesis.
Estimation of critical values in interacting particle systems
Authors: Gouet Raúl , López F. Javier , Sanz Gerardo
Published: July 2016

The estimation of critical values is one of the most interesting problems in the study of interacting particle systems. The bounds obtained analytically are not usually very tight and, therefore, computer simulation has been proved to be very useful in the estimation of these values. In this paper we present a new method for the estimation of critical values in any interacting particle system with an absorbing state. The method, based on the asymptotic behaviour of the absorption time of the process, is very easy to implement and provides good estimates. It can also be applied to processes different from particle systems.
A general minimal repair model
Authors: Aven Terje , Jensen Uwe
Published: July 2016

Minimal repairs have been given considerable attention in the reliability literature. Instead of replacing a failed system by a new one, such a minimal repair restores the system to the state it had just before failure. But the state just before failure depends on the information which is available about the system. Different information levels are possible. This paper gives a general definition characterizing point processes which describe time points of minimal repairs with respect to a certain information level. Some examples demonstrate the wide range of applications.
A join the shorter queue model in heavy traffic
Authors: Turner Stephen R. E.
Published: July 2016

We prove a new heavy traffic limit result for a simple queueing network under a ‘join the shorter queue’ policy, with the amount of traffic which has a routeing choice tending to zero as heavy traffic is approached. In this limit, the system considered does not exhibit state space collapse as in previous work by Foschini and Salz, and Reiman, but there is nevertheless some resource pooling gain over a policy of random routeing.
Distribution-free confidence intervals for measurement of effective bandwidth
Authors: Györfi László , Rácz András , Duffy Ken , Lewis John T. , Toomey Fergal
Published: July 2016

Hoeffding's inequality can be used in conjunction with the declared parameters of a traffic source, such as its peak rate, to obtain confidence intervals for measurements of the traffic's effective bandwidth. We describe a variety of interval-estimation procedures based on this idea, designed to provide differing degrees of robustness against non-stationarity. We also discuss how to compute confidence intervals for the effective bandwidth of an aggregate of traffic sources.
Quasi-renewal estimates
Authors: Piau Didier
Published: July 2016

We show that the solution of a quasi-renewal equation with an exponential distribution of the renewals converges at infinity and we compute explicitly the limit, hence generalizing the classical renewal theorem. We apply this result to a stochastic model of DNA replication introduced by Cowan and Chiu (1994).
A refinement of multivariate Bonferroni-type inequalities
Authors: Chen Tuhao , Seneta E.
Published: July 2016

Technology developed in a predecessor paper (Chen and Seneta (1996)) is applied to provide, in a unified manner, a sharpening of bivariate Bonferroni-type bounds on P(v
1≥r, v
2≥u) obtained by Galambos and Lee (1992; upper bound) and Chen and Seneta (1986; lower bound).
A note on uniformization for dynamic non-negative systems
Authors: van Dijk Nico M. , Sladký Karel
Published: July 2016

The classical technique of uniformization (or randomization) for bounded continuous-time Markov chains and Markov reward structures is extended to dynamic systems generated by arbitrary non-negative generators. Most notably, these include so-called input-output models in economic analysis. The results are of practical interest for both computational and theoretical purposes. Particularly, the recursive computation and the limiting behaviour of cumulative reward structures for non-negative dynamic systems is concluded as a special application. Two numerical examples are included to illustrate the conditions and the results.
Random and Poisson paced record models in the F α setup
Authors: Hofmann Glenn , Nagaraja H. N.
Published: July 2016

We study a random record model where the observationXihas continuous distribution functionFαi(αi> 0) and the number of available observations is random and independent of the observations. We obtain the joint distribution of the record values and inter-record times for our model. We investigate the distribution of the number of records when the number of observations has one of the common distributions and the α's increase geometrically or linearly. A particularly interesting case arises when the observations arrive at time points paced by a Poisson point process. For this model we obtain distributional results for the inter-arrival times of records for a large class of combinations of α structures and intensity functions.
Selecting a sequence of last successes in independent trials
Authors: Bruss F. Thomas , Paindaveine Davy
Published: July 2016

Let I
1,I
2,…,I

n
be a sequence of independent indicator functions defined on a probability space (Ω, A, P). We say that index k is a success time if I

k
= 1. The sequence I
1,I
2,…,I

n
is observed sequentially. The objective of this article is to predict the lth last success, if any, with maximum probability at the time of its occurrence. We find the optimal rule and discuss briefly an algorithm to compute it in an efficient way. This generalizes the result of Bruss (1998) for l = 1, and is equivalent to the problem of (multiple) stopping with l stops on the last l successes. We then extend the model to a larger class allowing for an unknown number N of indicator functions, and present, in particular, a convenient method for an approximate solution if the success probabilities are small. We also discuss some applications of the results.
Weak approximation for a class of Gaussian processes
Authors: Delgado Rosario , Jolis Maria
Published: July 2016

We prove that, under rather general conditions, the law of a continuous Gaussian process represented by a stochastic integral of a deterministic kernel, with respect to a standard Wiener process, can be weakly approximated by the law of some processes constructed from a standard Poisson process. An example of a Gaussian process to which this result applies is the fractional Brownian motion with any Hurst parameter.
Domain of attraction of the quasi-stationary distributions for the Ornstein-Uhlenbeck process
Authors: Lladser Manuel , San Martín Jaime
Published: July 2016

Let (Xt) be a one-dimensional Ornstein-Uhlenbeck process with initial density functionf: ℝ+→ ℝ+, which is a regularly varying function with exponent -(1 + η), η ∊ (0,1). We prove the existence of a probability measure ν with a Lebesgue density, depending on η, such that for everyA∊B(R+):
Oriented graphs generated by random points on a circle
Authors: Itoh Yoshiaki , Maehara Hiroshi , Tokushige Norihide
Published: July 2016

Extending the cascade model for food webs, we introduce a cyclic cascade model which is a random generation model of cyclic dominance relations. Put n species as n points Q
1,Q
2,…, Q

n
on a circle. If the counterclockwise way from Q

i
to Q

j
on the circle is shorter than the clockwise way, we say Q

i
dominates Q

j
. Consider a tournament whose dominance relations are generated from the points on a circle by this rule. We show that when we take n mutually independently distributed points on the circle, the probability of getting a regular tournament of order 2r+1 as the largest regular tournament is equal to (n/(2r+1))/2
n-1. This probability distribution is for the number of existing species after a sufficiently long period, assuming a Lotka-Volterra cyclic cascade model.
Quasistationarity in continuous-time Markov chains where absorption is not certain
Authors: Darlington S. J. , Pollett P. K.
Published: July 2016

In a recent paper [4] it was shown that, for an absorbing Markov chain where absorption is not guaranteed, the state probabilities at timetconditional on non-absorption bytgenerally depend ont. Conditions were derived under which there can be no initial distribution such that the conditional state probabilities are stationary. The purpose of this note is to show that these conditions can be relaxed completely: we prove, once and for all, that there arenocircumstances under which a quasistationary distribution can admit a stationary conditional interpretation.
On some waiting time problems
Authors: Stefanov Valeri T.
Published: July 2016

A unifying technology is introduced for finding explicit closed form expressions for joint moment generating functions of various random quantities associated with some waiting time problems. Sooner and later waiting times are covered for general discrete- and continuous-time models. The models are either Markov chains or semi-Markov processes with a finite number of states. Waiting times associated with generalized phase-type distributions, that are of interest in survival analysis and other areas, are also covered.
Reduction techniques for discrete-time Markov chains on totally ordered state space using stochastic comparisons
Authors: Truffet Laurent
Published: July 2016

We propose in this paper two methods to compute Markovian bounds for monotone functions of a discrete time homogeneous Markov chain evolving in a totally ordered state space. The main interest of such methods is to propose algorithms to simplify analysis of transient characteristics such as the output process of a queue, or sojourn time in a subset of states. Construction of bounds are based on two kinds of results: well-known results on stochastic comparison between Markov chains with the same state space; and the fact that in some cases a function of Markov chain is again a homogeneous Markov chain but with smaller state space. Indeed, computation of bounds uses knowledge on the whole initial model. However, only part of this data is necessary at each step of the algorithms.
A Monte Carlo approach to calculating probabilities for continuous identity by descent data
Authors: Browning Sharon
Published: July 2016

Two related individuals are identical by descent at a genetic locus if they share the same gene copy at that locus due to inheritance from a recent common ancestor. We consider idealized continuous identity by descent (IBD) data in which IBD status is known continuously along chromosomes. IBD data contains information about the relationship between the two individuals, and about the underlying crossover processes. We present a Monte Carlo method for calculating probabilities for IBD data. The method is not restricted to Haldane's Poisson process model of crossing-over but may be used with other models including the chi-square, Kosambi renewal and Sturt models. Results of a simulation study demonstrate that IBD data can be used to distinguish between alternative models for the crossover process.
Hazard rate and reversed hazard rate monotonicities in continuous-time Markov chains
Authors: Kijima Masaaki
Published: July 2016

A continuous-time Markov chain on the non-negative integers is called skip-free to the right (left) if only unit increments to the right (left) are permitted. If a Markov chain is skip-free both to the right and to the left, it is called a birth–death process. Karlin and McGregor (1959) showed that if a continuous-time Markov chain is monotone in the sense of likelihood ratio ordering then it must be an (extended) birth–death process. This paper proves that if an irreducible Markov chain in continuous time is monotone in the sense of hazard rate (reversed hazard rate) ordering then it must be skip-free to the right (left). A birth–death process is then characterized as a continuous-time Markov chain that is monotone in the sense of both hazard rate and reversed hazard rate orderings. As an application, the first-passage-time distributions of such Markov chains are also studied.
Negative binomial sums of random variables and discounted reward processes
Authors: Cooper William L.
Published: July 2016

Given a sequence of random variables (rewards), the Haviv–Puterman differential equation relates the expected infinite-horizon λ-discounted reward and the expected total reward up to a random time that is determined by an independent negative binomial random variable with parameters 2 and λ. This paper provides an interpretation of this proven, but previously unexplained, result. Furthermore, the interpretation is formalized into a new proof, which then yields new results for the general case where the rewards are accumulated up to a time determined by an independent negative binomial random variable with parameters k and λ.
Explicit sufficient invariants for an interacting particle system
Authors: Itoh Yoshiaki , Mallows Colin , Shepp Larry
Published: July 2016

We introduce a new class of interacting particle systems on a graph G. Suppose initially there are N

i
(0) particles at each vertex i of G, and that the particles interact to form a Markov chain: at each instant two particles are chosen at random, and if these are at adjacent vertices of G, one particle jumps to the other particle's vertex, each with probability 1/2. The process
N
enters a death state after a finite time when all the particles are in some independent subset of the vertices of G, i.e. a set of vertices with no edges between any two of them. The problem is to find the distribution of the death state, η
i
= N

i
(∞), as a function of N

i
(0).

We are able to obtain, for some special graphs, the limiting distribution of N

i
if the total number of particles N → ∞ in such a way that the fraction, N

i
(0)/S = ξ
i
, at each vertex is held fixed as N → ∞. In particular we can obtain the limit law for the graph S
2, the two-leaf star which has three vertices and two edges.
Heterogeneity in epidemic models and its effect on the spread of infection
Authors: Andersson Håkan , Britton Tom
Published: July 2016

We first study an epidemic amongst a population consisting of individuals with the same infectivity but with varying susceptibilities to the disease. The asymptotic final epidemic size is compared with the corresponding size for a homogeneous population. Then we group a heterogeneous population into households, assuming very high infectivity within households, and investigate how the global infection pressure is affected by rearranging individuals between the households. In both situations considered, it turns out that whether or not homogenizing the individuals or households will result in an increased spread of infection actually depends on the infectiousness of the disease.
A threshold limit theorem for the stochastic logistic epidemic
Authors: Andersson Håkan , Djehiche Boualem
Published: July 2016

The time until extinction for the closed SIS stochastic logistic epidemic model is investigated. We derive the asymptotic distribution for the extinction time as the population grows to infinity, under different initial conditions and for different values of the infection rate.
Optimal multivariate stopping rules
Authors: Assaf David , Samuel-Cahn Ester
Published: July 2016

For fixediletX(i) = (X1(i), …,Xd(i)) be ad-dimensional random vector with some known joint distribution. Hereishould be considered a time variable. LetX(i),i= 1, …,nbe a sequence ofnindependent vectors, wherenis the total horizon. In many examplesXj(i) can be thought of as the return to partnerj, when there ared≥ 2 partners, and one stops with theith observation. If thejth partner alone could decide on a (random) stopping rulet, his goal would be to maximizeEXj(t) over all possible stopping rulest≤n. In the present ‘multivariate’ setup thedpartners must however cooperate and stop at thesamestopping timet, so as to maximize some agreed functionh(∙) of the individual expected returns. The goal is thus to find a stopping rulet*for whichh(EX1(t), …,EXd(t)) =h(EX(t)) is maximized. For continuous and monotonehwe describe the class of optimal stopping rulest*. With some additional symmetry assumptions we show that the optimal rule is one which (also) maximizesEZtwhereZi= ∑dj=1Xj(i), and hence has a particularly simple structure. Examples are included, and the results are extended both to the infinite horizon case and to the case whenX(1), …,X(n) are dependent. Asymptotic comparisons between the present problem of finding suph(EX(t)) and the ‘classical’ problem of finding supEh(X(t)) are given. Comparisons between the optimal return to the statistician and to a ‘prophet’ are also included. In the present context a ‘prophet’ is someone who can base his (random) choicegon the full sequenceX(1), …,X(n), with corresponding return suph(EX(g)).
Inequalities for multinomial allocations with application to DNA fingerprinting
Authors: Grishechkin Sergei
Published: July 2016

We consider an allocation ofnballs intoNcells according to probabilitiespi. Assuming that the balls are allocated successively, denote by φ(n,N) the number of such balls which go into an already occupied cell. Ifn= 2 the probability that two balls will occupy the same cell is equal to the so-called match probabilityMP=p21+ … +p2N. An upper estimate for the probability ℙ(φ(n,N) ≤m) which depends only onnandMPis derived. Such inequalities are important for estimation of the reliability of DNA fingerprinting, a new method of crime investigation which is currently much debated.
Stochastic linearization: the theory
Authors: Bernard Pierre , Wu Liming
Published: July 2016

Very little is known about the quantitative behaviour of dynamical systems with random excitation, unless the system is linear. Known techniques imply the resolution of parabolic partial differential equations (Fokker–Planck–Kolmogorov equation), which are degenerate and of high dimension and for which there is no effective known method of resolution. Therefore, users (physicists, mechanical engineers) concerned with such systems have had to design global linearization techniques, known as equivalent statistical linearization (Roberts and Spanos (1990)). So far, there has been no rigorous justification of these techniques, with the notable exception of the paper by Kozin (1987). In this contribution, using large deviation principles, several mathematically founded linearization methods are proposed. These principles use relative entropy, or Kullback information, of two probability measures, and Donsker–Varadhan entropy of a Gaussian measure relatively to a Markov kernel. The method of ‘true linearization’ (Roberts and Spanos (1990)) is justified.
Superexponential decay for the GEM process
Authors: Zeitouni O.
Published: July 2016

We show that the GEM process has strong ordering properties: the probability that one of the k largest elements in the GEM sequence is beyond the first ck elements (c > 1) decays superexponentially in k.
Optimal stopping and maximal inequalities for geometric Brownian motion
Authors: Graversen S. E. , Peskir G.
Published: July 2016

Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem supτ
E (max0≤t≤τ
X

t
− c τ), where X = (X

t
)
t≥0 is geometric Brownian motion with drift μ and volatility σ > 0, and the supremum is taken over all stopping times for X. The payoff is shown to be finite, if and only if μ < 0. The optimal stopping time is given by τ* = inf {t > 0 | X

t
= g
* (max0≤t≤s

X

s
)} where s ↦ g
*(s) is the maximal solution of the (nonlinear) differential equation

under the condition 0 < g(s) < s, where Δ = 1 − 2μ / σ2 and K = Δ σ2 / 2c. The estimate is established g
*(s) ∼ ((Δ − 1) / K Δ)1 / Δ
s
1−1/Δ as s → ∞. Applying these results we prove the following maximal inequality:

where τ may be any stopping time for X. This extends the well-known identity E (sup
t>0
X

t
) = 1 − (σ 2 / 2 μ) and is shown to be sharp. The method of proof relies upon a smooth pasting guess (for the Stephan problem with moving boundary) and the Itô–Tanaka formula (being applied two-dimensionally). The key point and main novelty in our approach is the maximality principle for the moving boundary (the optimal stopping boundary is the maximal solution of the differential equation obtained by a smooth pasting guess). We think that this principle is by itself of theoretical and practical interest.
On the optimality of repair-cost-limit policies
Authors: Jiang Xiaoyue , Cheng Kan , Makis Viliam
Published: July 2016

An optimal repair/replacement problem for a single-unit repairable system with minimal repair and random repair cost is considered. The existence of the optimal policy is established using results of the optimal stopping theory, and it is shown that the optimal policy is a ‘repair-cost-limit’ policy, that is, there is a series of repair-cost-limit functionsgn(t),n= 1, 2,…, such that a unit of agetis replaced at thenth failure if and only if the repair costC(n,t) ≥gn(t); otherwise it is minimally repaired. If the repair cost does not depend onn, then there is a single repair cost limit functiong(t), which is uniquely determined by a first-order differential equation with a boundary condition.
One-dimensional loss networks and conditioned M/G/∞ queues
Authors: Ferrari Pablo A. , Lopes Garcia Nancy
Published: July 2016

We study one-dimensional continuous loss networks with length distribution G and cable capacity C. We prove that the unique stationary distribution η
L
of the network for which the restriction on the number of calls to be less than C is imposed only in the segment [−L,L] is the same as the distribution of a stationary M/G/∞ queue conditioned to be less than C in the time interval [−L,L]. For distributions G which are of phase type (= absorbing times of finite state Markov processes) we show that the limit as L → ∞ of η
L
exists and is unique. The limiting distribution turns out to be invariant for the infinite loss network. This was conjectured by Kelly (1991).
On critical vaccination coverage in multitype epidemics
Authors: Britton Tom
Published: July 2016

Consider an epidemic outbreak in a large population resulting in an overall proportion infected p~. The proportion needed to be vaccinated in order to prevent the epidemic, the critical vaccination coverage v
c, depends on individual and social heterogeneities in the population. In the present note it is shown that v
c is larger if, as is likely, individuals differ in terms of susceptibility than if they are equally susceptible.
The stress tensor and the energy of a continuous time homogeneous Markov system with fixed size
Authors: Tsaklidis George M.
Published: July 2016

The set of the attainable structures of a continuous time homogeneous Markov system (HMS) with fixed size, is considered as a continuum and the evolution of the HMS in the Euclidean space corresponds to its motion. Taking account of the velocity field of the HMS, a suitable model of continuum–defined by its stress tensor–is proposed in order to explain the motion of the system. The adoption of this model (equivalently of its stress tensor) enables us to establish the concept of the energy of a structure of the HMS.
Bounds for the total variation distance between the binomial and the Poisson distribution in case of medium-sized success probabilities
Authors: Weba Michael
Published: July 2016

In applied probability, the distribution of a sum of n independent Bernoulli random variables with success probabilities p
1,p
2,…, p

n
is often approximated by a Poisson distribution with parameter λ = p
1 + p
2 + p

n
. Popular bounds for the approximation error are excellent for small values, but less efficient for moderate values of p
1,p
2,…,p

n
.

Upper bounds for the total variation distance are established, improving conventional estimates if the success probabilities are of medium size. The results may be applied directly, e.g. to approximation problems in risk theory.
Asymptotic behavior of a multiplexer fed by a long-range dependent process
Authors: Liu Zhen , Nain Philippe , Towsley Don , Zhang Zhi-Li
Published: July 2016

In this paper we study the asymptotic behavior of the tail of the stationary backlog distribution in a single server queue with constant service capacity c, fed by the so-called M/G/∞ input process or Cox input process. Asymptotic lower bounds are obtained for any distribution G and asymptotic upper bounds are derived when G is a subexponential distribution. We find the bounds to be tight in some instances, e.g. when G corresponds to either the Pareto or lognormal distribution and c − ρ < 1, where ρ is the arrival rate at the buffer.
Continuity for multi-type branching processes with varying environments
Authors: Dafydd Jones Owen
Published: July 2016

Conditions are derived for the components of the normed limit of a multi-type branching process with varying environments, to be continuous on (0, ∞). The main tool is an inequality for the concentration function of sums of independent random variables, due originally to Petrov. Using this, we show that if there is a discontinuity present, then a particular linear combination of the population types must converge to a non-random constant (Equation (1)). Ensuring this can not happen provides the desired continuity conditions.
Using fuzzy sets in manpower planning
Authors: Guerry M. A.
Published: July 2016

In this paper, given personnel distributions that are not attainable, we introduce the grade of attainability in order to measure the degree to which there exists a similar distribution that is attainable. For constant size systems controlled by recruitment, properties of the most similar distribution to a given distribution are formulated.
Hedging in discrete time under transaction costs and continuous-time limit
Authors: Koehl Pierre-F. , Pham Huyên , Touzi Nizar
Published: July 2016

We consider a discrete-time financial market model with L
1 risky asset price process subject to proportional transaction costs. In this general setting, using a dual martingale representation we provide sufficient conditions for the super-replication cost to coincide with the replication cost. Next, we study the convergence problem in a stationary binomial model as the time step tends to zero, keeping the proportional transaction costs fixed. We derive lower and upper bounds for the limit of the super-replication cost. In the case of European call options and for a unit initial holding in the risky asset, the upper and lower bounds are equal. This result also holds for the replication cost of European call options. This is evidence (but not a proof) against the common opinion that the replication cost is infinite in a continuous-time model.
Exact distribution of word occurrences in a random sequence of letters
Authors: Robin S. , Daudin J. J.
Published: July 2016

The study of the distribution of the distance between words in a random sequence of letters is interesting in view of application in genome sequence analysis. In this paper we give the exact distribution probability and cumulative distribution function of the distances between two successive occurrences of a given word and between the nth and the (n+m)th occurrences under three models of generation of the letters: i.i.d. with the same probability for each letter, i.i.d. with different probabilities and Markov process. The generating function and the first two moments are also given. The point of studying the distances instead of the counting process is that we get some knowledge not only about the frequency of a word but also about its longitudinal distribution in the sequence.
Delay in polling systems with large switch-over times
Authors: van der Mei R. D.
Published: July 2016

We study the delay in cyclic polling systems with mixtures of gated and exhaustive service, and with deterministic switch-over times. We show that, under proper scalings, the waiting-time distribution at each of the queues converges to a uniform distribution over a known interval when the switch-over times tend to infinity.
Asymptotic behaviour for random walks in random environments
Authors: Alili S.
Published: July 2016

In this paper we consider limit theorems for a random walk in a random environment, (X

n
). Known results (recurrence-transience criteria, law of large numbers) in the case of independent environments are naturally extended to the case where the environments are only supposed to be stationary and ergodic. Furthermore, if ‘the fluctuations of the random transition probabilities around are small’, we show that there exists an invariant probability measure for ‘the environments seen from the position of (X

n
)’. In the case of uniquely ergodic (therefore non-independent) environments, this measure exists as soon as (X

n
) is transient so that the ‘slow diffusion phenomenon’ does not appear as it does in the independent case. Thus, under regularity conditions, we prove that, in this case, the random walk satisfies a central limit theorem for any fixed environment.
A laplace transform representation in a class of renewal queueing and risk processes
Authors: Willmot Gordon E.
Published: July 2016

For a class of renewal queueing processes characterized by a rational Laplace–Stieltjes transform of the arrival inter-occurrence time distribution, the Laplace–Stieltjes transform of the equilibrium (actual) waiting time distribution is re-expressed in a manner which facilitates explicit inversion under certain conditions. The results are of interest in other contexts as well, as for example in insurance ruin theory. Various analytic properties of these quantities are then obtained as a result.
A new stochastic model of microsatellite evolution
Authors: Durrett Richard , Kruglyak Semyon
Published: July 2016

We introduce a continuous-time Markov chain model for the evolution of microsatellites, simple sequence repeats in DNA. We prove the existence of a unique stationary distribution for our model, and fit the model to data from approximately 106base pairs of DNA from fruit flies, mice, and humans. The slippage rates from the best fit for our model are consistent with experimental findings.
Modelling hierarchical systems by a continuous-time homogeneous Markov chain using two-wave panel data
Authors: Carette Philippe
Published: July 2016

An open hierarchical (manpower) system divided into a totally ordered set of k grades is discussed. The transitions occur only from one grade to the next or to an additional (k+1)th grade representing the external environment of the system. The model used to describe the dynamics of the system is a continuous-time homogeneous Markov chain with k+1 states and infinitesimal generator
R
= (r

ij
) satisfying r

ij
= 0 if i > j or i + 1 < j ≤ k (i, j = 1,…,k+1), the transition matrix
P
between times 0 and 1 being
P
= exp
R
. In this paper, two-wave panel data about the hierarchical system are considered and the resulting fact that, in general, the maximum-likelihood estimated transition matrix cannot be written as an exponential of an infinitesimal generator
R
having the form described above. The purpose of this paper is to investigate when this can be ascribed to the effect of sampling variability.
On the probabilistic properties of a double threshold ARMA conditional heteroskedastic model
Authors: Ling Shiqing
Published: July 2016

Following Tweedie (1988), this paper constructs a special test function which leads to sufficient conditions for the stationarity and finiteness of the moments of a general non-linear time series model, the double threshold ARMA conditional heteroskedastic (DTARMACH) model. The results are applied to two well-known special cases, the GARCH and threshold ARMA (TARMA) models. The condition for the finiteness of the moments of the GARCH model is simple and easier to check than the condition given by Milhøj (1985) for the ARCH model. The condition for the stationarity of the TARMA model is identical to the condition given by Brockwell et al. (1992) for a special case, and verifies their conjecture that the moving average component does not affect the stationarity of the model. Under an additional irreducibility assumption, the geometric ergodicity of the GARCH and TARMA models is also established.
Large deviations for the time of ruin
Authors: Nyrhinen Harri
Published: July 2016

Let {Y

n
| n=1,2,…} be a stochastic process and M a positive real number. Define the time of ruin by T = inf{n | Y

n
> M} (T = +∞ if Y

n
≤ M for n=1,2,…). We are interested in the ruin probabilities for large M. Define the family of measures {P

M
| M > 0} by P

M
(B) = P(T/M ∊ B) for B ∊ ℬ (ℬ = Borel sets of ℝ). We prove that for a wide class of processes {Y

n
}, the family {P

M
} satisfies a large deviations principle. The rate function will correspond to the approximation P(T/M ≈ x) ≈ P(Y
⌈xM⌉/M ≈ 1) for x > 0. We apply the result to a simulation problem.
Poisson process via martingale and related characteristics
Authors: Wesołowski Jacek
Published: July 2016

The classical martingale characterizations of the Poisson process were obtained for point process or purely discontinuous martingale i.e. under additional assumptions on properties of trajectories. Here our aim is to search for related characterizations without relying on properties of trajectories. Except for a new martingale characterization, results based on conditional moments jointly involving the past and the nearest future are presented.
Bounded normal approximation in simulations of highly reliable Markovian systems
Authors: Tuffin Bruno
Published: July 2016

In this paper, we give necessary and sufficient conditions to ensure the validity of confidence intervals, based on the central limit theorem, in simulations of highly reliable Markovian systems. We resort to simulations because of the frequently huge state space in practical systems. So far the literature has focused on the property of bounded relative error. In this paper we focus on ‘bounded normal approximation’ which asserts that the approximation of the normal law, suggested by the central limit theorem, does not deteriorate as the reliability of the system increases. Here we see that the set of systems with bounded normal approximation is (strictly) included in the set of systems with bounded relative error.
Duality results for block-structured transition matrices
Authors: Zhao Yiqiang Q. , Li Wei , Alfa Attahiru Sule
Published: July 2016

In this paper, we consider a certain class of Markov renewal processes where the matrix of the transition kernel governing the Markov renewal process possesses some block-structured property, including repeating rows. Duality conditions and properties are obtained on two probabilistic measures which often play a key role in the analysis and computations of such a block-structured process. The method used here unifies two different concepts of duality. Applications of duality are also provided, including a characteristic theorem concerning recurrence and transience of a transition matrix with repeating rows and a batch arrival queueing model.
Computing approximate stationary distributions for discrete Markov processes with banded infinitesimal generators
Authors: Borges Carlos F. , Peters Craig S.
Published: July 2016

We develop an algorithm for computing approximations to the stationary distribution of a discrete birth-and-death process, provided that the infinitesimal generator is a banded matrix. We begin by computing stationary distributions for processes whose infinitesimal generators are Hessenberg. Our derivation in this special case is different from the classical case but it leads to the same result. We then show how to extend these ideas to processes where the infinitesimal generator is banded (or half-banded) and to quasi-birth–death processes. Finally, we give an example of the application of this method to a nearly completely decomposable Markov chain to demonstrate the general applicability of the technique.
The general coalescent with asynchronous mergers of ancestral lines
Authors: Sagitov Serik
Published: July 2016

Take a sample of individuals in the fixed-size population model with exchangeable family sizes. Follow the ancestral lines for the sampled individuals backwards in time to observe the ancestral process. We describe a class of asymptotic structures for the ancestral process via a convergence criterion. One of the basic conditions of the criterion prevents simultaneous mergers of ancestral lines. Another key condition implies that the marginal distribution of the family size is attracted by an infinitely divisible distribution. If the latter is normal the coalescent allows only for pairwise mergers (Kingman's coalescent). Otherwise multiple mergers happen with positive probability.
Reliability of a system with Poisson inspection times
Authors: Dieulle Laurence
Published: July 2016

For systems subject to inspections at Poisson random times, we present an analytic method which gives upper and lower bounds for the reliability. We also study its asymptotic behaviour and derive the asymptotic failure rate.
Minimizing response times and queue lengths in systems of parallel queues
Authors: Koole Ger , Sparaggis Panayotis D. , Towsley Don
Published: July 2016

We consider the problem of routeing customers to one of two parallel queues. Arrivals are independent of the state of the system but otherwise arbitrary. Assuming that queues have infinite capacities and the service times form a sequence of i.i.d. random variables with increasing likelihood ratio (ILR) distribution, we prove that the shortest queue (SQ) policy minimizes various cost functionals related to queue lengths and response times. We give a counterexample which shows that this result is not generally true when the service times have increasing hazard rate but are not increasing in the likelihood rate sense. Finally, we show that when capacities are finite the SQ policy stochastically maximizes the departure process and minimizes the loss counting process.
Asymptotic distribution for the sum and maximum of Gaussian processes
Authors: McCormick W. P. , Qi Y.
Published: July 2016

Previous work on the joint asymptotic distribution of the sum and maxima of Gaussian processes is extended here. In particular, it is shown that for a stationary sequence of standard normal random variables with correlation function r, the condition r(n)ln n = o(1) as n → ∞ suffices to establish the asymptotic independence of the sum and maximum.
Discounted optimal stopping problems for the maximum process
Authors: Pedersen Jesper Lund
Published: July 2016

The maximality principle [6] is shown to be valid in some examples of discounted optimal stopping problems for the maximum process. In each of these examples explicit formulas for the value functions are derived and the optimal stopping times are displayed. In particular, in the framework of the Black-Scholes model, the fair prices of two lookback options with infinite horizon are calculated. The main aim of the paper is to show that in each considered example the optimal stopping boundary satisfies the maximality principle and that the value function can be determined explicitly.
Sharp results on convergence rates for the distribution of GI/M/1/K queues as K tends to infinity
Authors: Choi Bong Dae , Kim Bara
Published: July 2016

In this paper, we investigate how fast the stationary distribution π
(K) of an embedded Markov chain (time-stationary distribution
q

(K) of the GI/M/1/K queue converges to the stationary distribution π of the embedded Markov chain (time-stationary distribution
q
) of the GI/M/1 queue as K tends to infinity. Simonot (1997) proved certain equalities. We obtain sharper results than these by finding limit values lim
K→∞σ-K
||π(K) - π|| and lim
K→∞σ-K
||
q

(K) -
q
|| explicitly.
Simple ratio prophet inequalities for a mortal with multiple choices
Authors: Assaf David , Samuel-Cahn Ester
Published: July 2016

Let X

i
≥ 0 be independent, i = 1,…, n, with known distributions and let X

n

*= max(X
1,…,X

n
). The classical ‘ratio prophet inequality’ compares the return to a prophet, which is EX

n

*, to that of a mortal, who observes the X

i
s sequentially, and must resort to a stopping rule t. The mortal's return is V(X
1,…,X

n
) = max EX

t
, where the maximum is over all stopping rules. The classical inequality states that EX

n

* < 2V(X
1,…,X

n
). In the present paper the mortal is given k ≥ 1 chances to choose. If he uses stopping rules t
1,…,t

k
his return is E(max(X

t
1,…,X

t

k
)). Let t(b) be the ‘simple threshold stopping rule’ defined to be the smallest i for which X

i
≥ b, or n if there is no such i. We show that there always exists a proper choice of k thresholds, such that EX

n

* ≤ ((k+1)/k)Emax(X

t
1,…,X

t

k
)), where t

i
is of the form t(b

i
) with some added randomization. Actually the thresholds can be taken to be thej/(k+1) percentile points of the distribution of X

n

*, j = 1,…,k, and hence only knowledge of the distribution of X

n

* is needed.