bluuebunny/crossref_metadata_2025 huggingface dataset:
Spatial epidemics with large finite range
Authors: Penrose Mathew D.
Published: July 2016

In the epidemic with removal with range r, each site z
, once infected, remains so for a period of time Tz
, the variables Tz
being i.i.d. with mean μ. While infected, a site infects its healthy r-neighbours independently at total rate α. After infection, sites become immune. We show that the critical rate of infection αc (r), above which an epidemic starting from a single site may continue forever, converges to μ
–1 as r →∞.
General branching processes with immigration
Authors: Olofsson Peter
Published: July 2016

A general multi-type branching process where new individuals immigrate according to some point process is considered. An intrinsic submartingale is defined and a convergence result for processes counted with random characteristics is obtained. Some examples are given.
Characterization of convergence rates for the approximation of the stationary distribution of infinite monotone stochastic matrices
Authors: Simonot F. , Song Y. Q.
Published: July 2016

Let P be an infinite irreducible stochastic matrix, recurrent positive and stochastically monotone and Pn
be any n × n stochastic matrix with Pn
≧ Tn
, where Tn
denotes the n × n northwest corner truncation of P. These assumptions imply the existence of limit distributions π and π
n
for P and Pn
respectively. We show that if the Markov chain with transition probability matrix P meets the further condition of geometric recurrence then the exact convergence rate of π
n
to π can be expressed in terms of the radius of convergence of the generating function of π. As an application of the preceding result, we deal with the random walk on a half line and prove that the assumption of geometric recurrence can be relaxed. We also show that if the i.i.d. input sequence (A(m)) is such that we can find a real number r
0 > 1 with , then the exact convergence rate of π
n
to π is characterized by r
0. Moreover, when the generating function of A is not defined for |z| > 1, we derive an upper bound for the distance between π
n
and π based on the moments of A.
Reward processes with nonlinear reward functions
Authors: Soltani A. Reza
Published: July 2016

Based on a semi-Markov process J(t), t ≧ 0, a reward process Z(t), t ≧ 0, is introduced where it is assumed that the reward function, p(k, x) is nonlinear; if the reward function is linear, i.e. ρ (k, x) = kx, the reward process Z(t), t ≧ 0, becomes the classical one, which has been considered by many authors. An explicit formula for E(Z(t)) is given in terms of the moments of the sojourn time distribution at t, when the reward function is a polynomial.
Forward and reversed time prediction of autoregressive sequences
Authors: Cambanis Stamatis , Fakhre-Zakeri Issa
Published: July 2016

Prediction for autoregressive sequences with finite second moment and of general order is considered. It is shown that the best predictor with time reversed is linear if and only if the innovations are Gaussian. The connection to time reversibility is also discussed.
Functional limit theory for the spectral covariance estimator
Authors: Dehay Dominique , Leśkow Jacek
Published: July 2016

Processes that exhibit repeatability in their kth-order moments are frequently studied in signal analysis. Such repeatability can be conveniently expressed with the help of almost periodicity. In particular, almost periodically correlated (APC) processes play an important role in the analysis of repeatable signals. This paper presents a study of asymptotic distributions of the estimator of the spectral covariance function for APC processes. It is demonstrated that, for a large class of APC processes, the functional central limit theorem holds.
Convergence rates for M/G/1 queues and ruin problems with heavy tails
Authors: Asmussen Søren , Teugels Jozef L.
Published: July 2016

The time-dependent virtual waiting time in a M/G/1 queue converges to a proper limit when the traffic intensity is less than one. In this paper we give precise rates on the speed of this convergence when the service time distribution has a heavy regularly varying tail.

The result also applies to the classical ruin problem. We obtain the exact rate of convergence for the ruin probability after time t for the case where claims arrive according to a Poisson process and claim sizes are heavy tailed.

Our result supplements similar theorems on exponential convergence rates for relaxation times in queueing theory and ruin probabilities in risk theory.
An extended class of time-continuous branching processes
Authors: Chen Rong-Rong
Published: July 2016

This paper is devoted to studying an extended class of time-continuous branching processes, motivated by the study of stochastic control theory and interacting particle systems. The uniqueness, extinction, recurrence and positive recurrence criteria for the processes are presented. The main new point in our proofs is the use of several different comparison methods. The resulting picture shows that the methods are effective and hence should also be meaningful in other situations.
A comparison of convergence rates for three models in the theory of dams
Authors: Lund Robert , Smith Walter
Published: July 2016

This paper compares the convergence rate properties of three storage models (dams) driven by time-homogeneous jump process input: the infinitely high dam, the finite dam, and the infinitely deep dam. We show that the convergence rate of the infinitely high dam depends on the moment properties of the input process, the finite dam always approaches its limiting distribution exponentially fast, and the infinitely deep dam approaches its limiting distribution exponentially fast under very general conditions. Our methods make use of rate results for regenerative processes and several sample path orderings.
Two-person red-and-black stochastic games
Authors: Secchi Piercesare
Published: July 2016

We define a leavable stochastic game which is a possible two-person generalization of the classical red-and-black gambling problem. We show that there are three basic possibilities for a two-person red-and-black game which, by analogy with gambling theory, we call the subfair, the fair and the superfair cases. A suitable generalization of what in gambling theory is called bold play is proved to be a uniformly ε-optimal stationary strategy for player I in the fair and the subfair cases whereas a generalization of timid play is shown to be ε-optimal for player I in the superfair possibility.
Analysis of a two-queue model with Bernoulli schedules
Authors: Lee Duan-Shin
Published: July 2016

In this paper we analyze a single server two-queue model with Bernoulli schedules. This discipline is very flexible and contains the exhaustive and 1-limited disciplines as special cases. We formulate the queueing system as a Riemann boundary value problem with shift. The boundary value problem is solved by exploring a Fredholm integral equation around the unit circle. Some numerical examples are presented at the end of the paper.
Steady state solution of a single-server queue with linear repeated requests
Authors: Artalejo J. R. , Gomez-Corral A.
Published: July 2016

Queueing systems with repeated requests have many useful applications in communications and computer systems modeling. In the majority of previous work the repeat requests are made individually by each unsatisfied customer. However, there is in the literature another type of queueing situation, in which the time between two successive repeated attempts is independent of the number of customers applying for service. This paper deals with the M/G/1 queue with repeated orders in its most general setting, allowing the simultaneous presence of both types of repeat requests. We first study the steady state distribution and the partial generating functions. When the service time distribution is exponential we show that the performance characteristics can be expressed in terms of hypergeometric functions.
Tandem queues with bulk arrivals, infinitely many servers and correlated service times
Authors: Sumita Ushio , Masuda Yasushi
Published: July 2016

A system of GIx
/G/∞ queues in tandem is considered where the service times of a customer are correlated but the service time vectors for customers are independently and identically distributed. It is shown that the binomial moments of the joint occupancy distribution can be generated by a sequence of renewal equations. The distribution of the joint occupancy level is then expressed in terms of the binomial moments. Numerical experiments for a two-station tandem queueing system demonstrate a somewhat counterintuitive result that the transient covariance of the joint occupancy level decreases as the covariance of the service times increases. It is also shown that the analysis is valid for a network of GIx/SM/∞ queues.
An invariant-sum characterization of Benford's law
Authors: Allaart Pieter C.
Published: July 2016

The accountant Nigrini remarked that in tables of data distributed according to Benford's law, the sum of all elements with first digit d (d = 1, 2,· ··, 9) is approximately constant. In this note, a mathematical formulation of Nigrini's observation is given and it is shown that Benford's law is the unique probability distribution such that the expected sum of all elements with first digits d1, · ··, dk
is constant for every fixed k.
A unique representation theorem for the conditional expectation of stationary processes and application to dynamic estimation problems
Authors: Campi Marco
Published: July 2016

In this paper, multivariate strict sense stationary stochastic processes are considered. It is shown that there exists a universal function by means of which the conditional expectation of any stationary process with respect to its past can be represented. This requires no ergodicity assumptions. The important implications of this result in the evaluation of the achievable performance in certain dynamic estimation problems with incomplete statistical information are also discussed.
Monotonicity results for MR/GI/1 queues
Authors: Bäuerle Nicole
Published: July 2016

This paper considers queues with a Markov renewal arrival process and a particular transition matrix for the underlying Markov chain. We study the effect that the transition matrix has on the waiting time of the nth customer as well as on the stationary waiting time. The main theorem generalizes results of Szekli et al. (1994a) and partly confirms their conjecture. In this context we show the importance of a new stochastic ordering concept.
Moment inequalities for sums of DMRL random variables
Authors: Fagiuoli Enrico , Pellerey Franco
Published: July 2016

Some moment inequalities are known to be valid for non-parametric lifetime distribution classes. Here we consider one set of these inequalities, which hold for random variables that are DMRL (decreasing in mean residual life). We prove that such inequalities are satisfied by variables which are sums of DMRL random variables too, though these sums are not necessarily DMRL. Related results are shown, together with similar results valid for the stochastic comparison in mean residual life.
Comparison of system reliability functions under laboratory and common operating environments
Authors: Woodham Sara-Anne , Richards Donald St. P.
Published: July 2016

We compare R
1(t), the reliability function of a redundant m-of-n system operating within the laboratory, with R
D(t), the reliability function of the same system operating subject to environmental effects. Within the laboratory, all component lifetimes are independent and identically distributed according to G(α + 1, λ), a gamma distribution with index α + 1 and scale λ. Outside the laboratory, we adopt the model of Lindley and Singpurwalla (J. Appl. Prob. 23 (1986), 418-431) and assume that, conditional on a positive random variable η which models the effect of the common environment, all component lifetimes are independent and identically distributed according to G(α + 1, λη). When α is a non-negative integer we prove that for R
D(t) to underestimate (resp. overestimate) R
1(t) for all t sufficiently close to zero, it is necessary and sufficient that E(η
(n-m + 1)(α+1)) > 1 (resp. E(η
(n-m + 1)(α+1)) < 1). In the case in which n = 2, m= 1 and α = 0 we obtain a special case of a result of Currit and Singpurwalla (J. Appl. Prob. 26 (1988), 763-771). As an application, we obtain a necessary and sufficient condition under which R
D(t) initially understimates (or overestimates) R
1(t) when η follows a gamma or an inverse Gaussian distribution.
Filtering of a branching process given its split times
Authors: Ceci Claudia , Gerardi Anna
Published: July 2016

A filtering problem for a branching process is studied within the tree framework, and the minimal intensity of the splitting process is given.
First-passage-time densities for time-non-homogeneous diffusion processes
Authors: Gutiérrez R. , Ricciardi L. M. , Román P. , Torres F.
Published: July 2016

In this paper we study a Volterra integral equation of the second kind, including two arbitrary continuous functions, in order to determine first-passage-time probability density functions through time-dependent boundaries for time-non-homogeneous one-dimensional diffusion processes with natural boundaries. These results generalize those which were obtained for time-homogeneous diffusion processes by Giorno et al. [3], and for some particular classes of time-non-homogeneous diffusion processes by Gutiérrez et al. [4], [5].
Average case behavior of random search for the maximum
Authors: Calvin James M. , Glynn Peter W.
Published: July 2016

This paper is a study of the error in approximating the global maximum of a Brownian motion on the unit interval by observing the value at randomly chosen points. One point of view is to look at the error from random sampling for a given fixed Brownian sample path; another is to look at the error with both the path and observations random. In the first case we show that for almost all Brownian paths the error, normalized by multiplying by the square root of the number of observations, does not converge in distribution, while in the second case the normalized error does converge in distribution. We derive the limiting distribution of the normalized error averaged over all paths.
Regularity and reversibility results for birth-death-migration processes
Authors: Renshaw Eric , Dai Yonglong
Published: July 2016

A spatial process is considered in which two general birth-death processes are linked by migration of individuals. We examine conditions for weak symmetry and regularity, and develop necessary and sufficient conditions for recurrence. The results are easily extended to the k-process case.
Limit theorems for the total size of a spatial epidemic
Authors: Andersson Håkan , Djehiche Boualem
Published: July 2016

We study the long-term behaviour of a sequence of multitype general stochastic epidemics, converging in probability to a deterministic spatial epidemic model, proposed by D. G. Kendall. More precisely, we use branching and deterministic approximations in order to study the asymptotic behaviour of the total size of the epidemics as the number of types and the number of individuals of each type both grow to infinity.
Characterization of life distributions under some generalized stochastic orderings
Authors: Cai Jun , Wu Yanhong
Published: July 2016

In this paper we investigate the characterizations of life distributions under four stochastic orderings, <
p
, < (p), < (p) and < L, by a unified method. Conditions for the stochastic equality of two non-negative random variables under the four stochastic orderings are derived. Many previous results are consequences. As applications, we provide characterizations of life distributions by a single value of their Laplace transforms under orderings <
p
and < (p) and their moment generating functions under orderings < p
and < (p). Under ordering < L, a characterization is given by the expected value of a strictly completely monotone function. The conditions for the stochastic equality of two non-negative vectors under the stochastic orderings < (p), < (p) and < L are presented in terms of the Laplace transforms and moment generating functions of their extremes and sample means. Characterizations of the exponential distribution among L and L life distribution classes are also given.
On the move-to-front scheme with Markov dependent requests
Authors: Phatarfod R. M. , Pryde A. J. , Dyte David
Published: July 2016

In this paper we consider the operation of the move-to-front scheme where the requests form a Markov chain of N states with transition probability matrix P
. It is shown that the configurations of items at successive requests form a Markov chain, and its transition probability matrix has eigenvalues that are the eigenvalues of all the principal submatrices of P except those of order N—1. We also show that the multiplicity of the eigenvalues of submatrices of order m is the number of derangements of N — m objects. The last result is shown to be true even if P is not a stochastic matrix.
The geometric convergence rate of a Lindley random walk
Authors: Lund Robert B.
Published: July 2016

Let {Xn
} be the Lindley random walk on [0,∞) defined by . Here, {An
} is a sequence of independent and identically distributed random variables. When for some r > 1, {Xn
} converges at a geometric rate in total variation to an invariant distribution π; i.e. there exists s > 1 such that for every initial state . In this communication we supply a short proof and some extensions of a large deviations result initially due to Veraverbeke and Teugels (1975, 1976): the largest s satisfying the above relationship is and satisfies φ ‘(r
0) = 0.
Two chain-transformations and their applications to quantiles
Authors: Bertoin J. , Chaumont L. , Yor M.
Published: July 2016

We describe two chain-transformations which explain and extend identities for order statistics and quantiles proved by Wendel, Port and, more recently, by Dassios.
Multiple buying or selling with vector offers
Authors: Bruss F. Thomas , Ferguson Thomas S.
Published: July 2016

We consider a generalization of the house-selling problem to sellingkhouses. Let the offers,X1,X2, · ··,be independent, identically distributedk-dimensional random vectors having a known distribution with finite second moments. The decision maker is to choose simultaneouslykstopping rules,N1, · ··,Nk,one for each component. The payoff is the sum overjof thejth component ofminus a constant cost per observation until all stopping rules have stopped. Simple descriptions of the optimal rules are found. Extension is made to problems with recall of past offers and to problems with a discount.
Branching processes and functional differential equations determining steady-size distributions in cell populations
Authors: Taib Ziad
Published: July 2016

The functional differential equation y′(x) = ay(λx) + by(x) arises in many different situations. The purpose of this note is to show how it arises in some multitype branching process cell population models. We also show how its solution can be given an intuitive interpretation as the probability density function of an infinite sum of independent but not identically distributed random variables.
New partial ordering of survival functions based on the notion of uncertainty
Authors: Ebrahimi Nader , Pellerey Franco
Published: July 2016

A new partial ordering among life distributions in terms of their uncertainties is introduced. Our measure of uncertainty is Shannon information applied to the residual lifetime. The relationship between this ordering and various existing orderings of life distributions are discussed. Various properties of our proposed concept are examined. Based on our proposed ordering and various existing orderings, the notion of a ‘better system' is introduced.
A limit theorem for the replication time of a DNA molecule
Authors: Cowan Richard , Chiu S. N. , Holst Lars
Published: July 2016

The DNA of higher animals replicates by an interesting mechanism. Enzymes recognise specific sites randomly scattered on the molecule and establish a bidirectional process of unwinding and replication from these sites. We investigate the limiting distribution of the completion time for this process by considering related coverage problems investigated by Janson (1983) and Hall (1988).
Problèmes de premier passage résolubles par la méthode de séparation des variables
Authors: Lefebvre Mario
Published: July 2016

In this paper, bidimensional stochastic processes defined by ax(t) = y(t)dt and dy(t) = m(y)dt + [2v(y)]1/2
dW(t), where W(t) is a standard Brownian motion, are considered. In the first section, results are obtained that allow us to characterize the moment-generating function of first-passage times for processes of this type. In Sections 2 and 5, functions are computed, first by fixing the values of the infinitesimal parameters m(y) and v(y) then by the boundary of the stopping region.
Optimal software release policy based on a two-person game of timing
Authors: Zeephongsekul P. , Chiera C.
Published: July 2016

This paper presents a software release policy based on a two-person game of timing. Existing release policies depend solely on cost factors and ignore the element of competition between rival producers, whereas in our policy both of these factors are taken into consideration. Through a series of preliminary results, it is shown that an optimal release policy exists as a Nash equilibrium point in the space of mixed strategies. We also present numerical examples of this optimal policy applied to software reliability growth models which are based on the non-homogeneous Poisson process.
Limiting behavior of some measures of system availability
Authors: Mi Jie
Published: July 2016

Availability is an important characteristic of a system. Different types of availability are defined. For the case when a sequence of bivariate random variables of lifetime and repair time are i.i.d. certain properties have been established previously. In practice, however, we need to consider the situation where these bivariate random variables are independent but not identically distributed. Properties of two measures of availability for the i.i.d. case are extended to this more general case.
Conditions for integrability of Markov chains
Authors: Hamza K. , Klebaner F. C.
Published: July 2016

We give simple sufficient conditions for integrability of continuous-time Markov chains in terms of their infinitesimal parameters. Similar conditions for regularity are stated first, and a simple proof given.
Random perturbations of recursive sequences with an application to an epidemic model
Authors: Viaud Daniel Pierre Loti
Published: July 2016

We investigate the asymptotic sample path behaviour of a randomly perturbed discrete-time dynamical system. We consider the case where the trajectories of the non-perturbed dynamical system are attracted by a finite number of limit sets and characterize a case where this property remains valid for the perturbed dynamical system when the perturbation converges to zero. For this purpose, no further assumptions on the perturbation are needed and our main condition applies to the limit sets of the non-perturbed dynamical system. When the limit sets reduce to limit points we show that this main condition is more general than the usual assumption of the existence of a Lyapunov function for the non-perturbed dynamical system. An application to an epidemic model is given to illustrate our results.
A note on the Volterra integral equation for the first-passage-time probability density
Authors: Jáimez R. Gutiérrez , Román P. Román , Ruiz F. Torres
Published: July 2016

In this paper we prove the validity of the Volterra integral equation for the evaluation of first-passage-time probability densities through varying boundaries, given by Buonocore et al. [1], for the case of diffusion processes not necessarily time-homogeneous. We study, specifically those processes that can be obtained from the Wiener process in the sense of [5]. A study of the kernel of the integral equation, in the same way as that by Buonocore et al. [1], is done. We obtain the boundaries for which closed-form solutions of the integral equation, without having to solve the equation, can be obtained. Finally, a few examples are given to indicate the actual use of our method.
Ruin probabilities via local adjustment coefficients
Authors: Asmussen Søren , Nielsen Hanne Mandrup
Published: July 2016

Let ψ(u) be the ruin probability in a risk process with initial reserve u, Poisson arrival rate β, claim size distribution B and premium rate p(x) at level x of the reserve. Let y(x) be the non-zero solution of the local Lundberg equation . It is shown that is non-decreasing and that log ψ(u) ≈ –I(u) in a slow Markov walk limit. Though the results and conditions are of large deviations type, the proofs are elementary and utilize piecewise comparisons with standard risk processes with a constant p. Also simulation via importance sampling using local exponential change of measure defined in terms of the γ(x) is discussed and some numerical results are presented.
Convergence to stationary state for a Markov move-to-front scheme
Authors: Rodrigues Eliane R.
Published: July 2016

This work considers items (e.g. books, files) arranged in an array (e.g. shelf, tape) with N positions and assumes that items are requested according to a Markov chain (possibly, of higher order). After use, the requested item is returned to the leftmost position of the array. Successive applications of the procedure above give rise to a Markov chain on permutations. For equally likely items, the number of requests that makes this Markov chain close to its stationary state is estimated. To achieve that, a coupling argument and the total variation distance are used. Finally, for non-equally likely items and so-called p-correlated requests, the coupling time is presented as a function of the coupling time when requests are independent.
Reliability bounds for the ℒ-class and Laplace order
Authors: Sengupta Debasis
Published: July 2016

Sharp upper and lower reliability bounds for the ℒ- and -classes of life distributions with a given mean are provided. Bounds are also given for a distribution which is related to a known distribution (not necessarily exponential) through the Laplace order.
Stochastic vector difference equations with stationary coefficients
Authors: Glasserman Paul , Yao David D.
Published: July 2016

We give a unified presentation of stability results for stochastic vector difference equations based on various choices of binary operations and , assuming that are stationary and ergodic. In the scalar case, under standard addition and multiplication, the key condition for stability is
E
[log |A
0|] < 0. In the generalizations, the condition takes the form γ< 0, where γis the limit of a subadditive process associated with . Under this and mild additional conditions, the process has a unique finite stationary distribution to which it converges from all initial conditions.

The variants of standard matrix algebra we consider replace the operations + and × with (max, +), (max,×), (min, +), or (min,×). In each case, the appropriate stability condition parallels that for the standard recursions, involving certain subadditive limits. Since these limits are difficult to evaluate, we provide bounds, thus giving alternative, computable conditions for stability.
Compound Poisson approximations for word patterns under Markovian hypotheses
Authors: Geske Mark X. , Godbole Anant P. , Schaffner Andrew A. , Skolnick Allison M. , Wallstrom Garrick L.
Published: July 2016

Consider a stationary Markov chainwith state space consisting of theξ-letter alphabet set Λ= {a1, a2, ···,aξ}.We study the variablesM=M(n, k) andN=N(n, k),defined, respectively, as the number of overlapping and non-overlapping occurrences of a fixed periodick-letter word, and use the Stein–Chen method to obtain compound Poisson approximations for their distribution.
A decomposition theorem for infinite stochastic matrices
Authors: Heyman Daniel P.
Published: July 2016

We prove that every infinite-state stochastic matrix P say, that is irreducible and consists of positive-recurrrent states can be represented in the form I – P=(A – I)(B – S), where A is strictly upper-triangular, B is strictly lower-triangular, and S is diagonal. Moreover, the elements of A are expected values of random variables that we will specify, and the elements of B and S are probabilities of events that we will specify. The decomposition can be used to obtain steady-state probabilities, mean first-passage-times and the fundamental matrix.
On stochastic ordering of random vectors
Authors: Cohen Arthur , Sackrowitz Harold B.
Published: July 2016

Conditions that have been widely used to establish stochastic ordering of random vectors include those given by Kamae et al. (1977), Barlow and Proschan (1975) and Langberg (1988). In this note we offer conditions that are weaker than those above, thus providing potential application to a wider class of problems.
Recurring-with-carry sequences
Authors: Koç Cemal
Published: July 2016

In this note we introduce recurring-with-carry sequences which generalize add-with-carry and subtract-with-borrow sequences introduced in [1], and describe periods of admissible recurring-with-carry sequences which include add-with-carry and subtract-with-borrow sequences with a few exceptions.
Invariance principle for the deviation between the probability content and the interior point proportion of a random convex hull
Authors: Massé Bruno
Published: July 2016

The quadratic mean of the deviation between the probability content and the interior point proportion of a random convex hull inis investigated. We obtain, in particular, an explicit and distribution-independent bound.
On a conjecture for the non-existence of the expectation of randomly stopped sums
Authors: Lefèvre Claude , Utev Sergey
Published: July 2016

In a recent paper on the validity of Wald's equation, Roters (1994) raised an important question on the non-existence of the expectation of randomly stopped sums. The purpose of this note is to answer the question in the affirmative. As a consequence, an old question by Taylor (1972) also gets a positive answer.
Multivariate normal approximations by Stein's method and size bias couplings
Authors: Goldstein Larry , Rinott Yosef
Published: July 2016

Stein's method is used to obtain two theorems on multivariate normal approximation. Our main theorem, Theorem 1.2, provides a bound on the distance to normality for any non-negative random vector. Theorem 1.2 requires multivariate size bias coupling, which we discuss in studying the approximation of distributions of sums of dependent random vectors. In the univariate case, we briefly illustrate this approach for certain sums of nonlinear functions of multivariate normal variables. As a second illustration, we show that the multivariate distribution counting the number of vertices with given degrees in certain random graphs is asymptotically multivariate normal and obtain a bound on the rate of convergence. Both examples demonstrate that this approach may be suitable for situations involving non-local dependence. We also present Theorem 1.4 for sums of vectors having a local type of dependence. We apply this theorem to obtain a multivariate normal approximation for the distribution of the random p-vector, which counts the number of edges in a fixed graph both of whose vertices have the same given color when each vertex is colored by one of p colors independently. All normal approximation results presented here do not require an ordering of the summands related to the dependence structure. This is in contrast to hypotheses of classical central limit theorems and examples, which involve for example, martingale, Markov chain or various mixing assumptions.
Recursive relations for the distribution of waiting times in a Markov chain
Authors: Banjevic Dragan
Published: July 2016

A Markov chain is used as a model for a sequence of random experiments. The waiting time for sequence patterns is considered. Recursive-type relations for the distribution of waiting times are obtained.
Expansions for Markov-modulated systems and approximations of ruin probability
Authors: Błaszczyszyn Bartłomiej , Rolski Tomasz
Published: July 2016

Let N be a stationary Markov-modulated marked point process on ℝ with intensity β
∗ and consider a real-valued functional ψ(N). In this paper we study expansions of the form
Eψ(N) = a
0 + β
∗
a
1 + ·· ·+ (β∗
)
nan
+ o((β
∗)
n
) for β
∗→ 0. Formulas for the coefficients ai
are derived in terms of factorial moment measures of N. We compute a
1 and a
2 for the probability of ruin φ u
with initial capital u for the risk process in the Markov-modulated environment; a
0 = 0. Moreover, we give a sufficient condition for ϕu
to be an analytic function of β
∗. We allow the premium rate function p(x) to depend on the actual risk reserve.
Bonferroni inequalities and deviations of discrete distributions
Authors: Móri Tamás F.
Published: July 2016

In the paper we first show how to convert a generalized Bonferroni-type inequality into an estimation for the generating function of the number of occurring events, then we give estimates for the deviation of two discrete probability distributions in terms of the maximum distance between their generating functions over the interval [0, 1].
Performance analysis of the discrete-time GI/Geom/1/N queue
Authors: Chaudhry M. L. , Gupta U. C.
Published: July 2016

This paper presents an analysis of the single-server discrete-time finite-buffer queue with general interarrival and geometric service time,GI/Geom/1/N. Using the supplementary variable technique, and considering the remaining interarrival time as a supplementary variable, two variations of this model, namely the late arrival system with delayed access (LAS-DA) and early arrival system (EAS), have been examined. For both cases, steady-state distributions for outside observers as well as at random and prearrival epochs have been obtained. The waiting time analysis has also been carried out. Results for theGeom/G/1/Nqueue with LAS-DA have been obtained from theGI/Geom/1/Nqueue with EAS. We also give various performance measures. An algorithm for computing state probabilities is given in an appendix.
Asymptotic analysis of queueing systems with identical service
Authors: Karpelevitch F. I. , Kreinin A. Ya.
Published: July 2016

We consider a heavy traffic regime in queueing systems with identical service. These systems belong to the class of multi-phase systems with dependent service times in different service nodes. We study the limit behaviour of the waiting time vector in heavy traffic. Both transient behaviour and the stationary regime are considered. Our analysis is based on the conception of ‘approximated functionals', which appeared to be fruitful in weak convergence theory of stochastic processes.
The maximum and time to absorption of a left-continuous random walk
Authors: Green P. J.
Published: July 2016

For a left-continuous random walk, absorbing at 0, the joint distribution of the maximum and time to absorption is derived. A description of the tails of the distributions and a conditional limit theorem are obtained for the cases where absorption is certain.
Asymptotic probabilities in a critical age-dependent branching process
Authors: Weiner Howard J.
Published: July 2016

Let Z(t) denote the number of cells alive at time t in a critical Bellman-Harris age-dependent branching process, that is, where the mean number of offspring per parent is one. A comparison method is used to show for k ≧ 1, and a high-order moment condition on G(t), where G(t) is the cell lifetime distribution, that lim
t→∞
t
2
P[Z(t) = k] = ak
> 0, where {ak
} are constants.

The method is also applied to the total progeny in the critical process.
Order statistics, Poisson processes and repairable systems
Authors: Miller Douglas R.
Published: July 2016

Necessary and sufficient conditions are presented under which the point processes equivalent to order statistics of n i.i.d. random variables or superpositions of n i.i.d. renewal processes converge to a non-degenerate limiting process as n approaches infinity. The limiting process must be one of three types of non-homogeneous Poisson process, one of which is the Weibull process. These point processes occur as failure-time models in the reliability theory of repairable systems.
On joint exchangeability and conservative processes with stochastic rates
Authors: Saunders Roy
Published: July 2016

In a previous article Saunders (1975) investigated the form of transition probabilities for a generalization of conservative processes in which the usual transition rate parameters are replaced by time-dependent stochastic variables. The results of that investigation are given in terms of properties of exchangeable random variables and require that the process be in a particular initial state at time zero. This article removes the restriction on the initial state by using some properties of two sequences of jointly exchangeable variables. General results analogous to those obtained previously are shown to hold for general initial states.
An upper bound for the probability of a union
Authors: Hunter David
Published: July 2016

The problem of bounding P(∪ Ai
) given P(A
i) and P(A
i
A
j) for i ≠ j = 1, …, k goes back to Boole (1854) and Bonferroni (1936). In this paper a new family of upper bounds is derived using results in graph theory. This family contains the bound of Kounias (1968), and the smallest upper bound in the family for a given application is easily derivable via the minimal spanning tree algorithm of Kruskal (1956). The properties of the algorithm and of the multivariate normal and t distributions are shown to provide considerable simplifications when approximating tail probabilities of maxima from these distributions.
Mean recurrence times for k successes within m trials
Authors: Huntington Raymond J.
Published: July 2016

Given an unlimited sequence of Bernoulli trials, let E be the recurrent event that k or more successes occur within m consecutive trials in the sequence. Let μk, m
denote the mean recurrence time for E. Previous work finds expressions for μk, m
for limited values of k and m. The present paper derives a closed expression for μk, m
for all k, m. Tables for μk, m
are also presented.
Fractional integrals of stationary processes and the central limit theorem
Authors: Rosenblatt M.
Published: July 2016

A class of limit theorems involving asymptotic normality is derived for stationary processes whose spectral density has a singular behavior near frequency zero. Generally these processes have ‘long-range dependence’ but are generated from strongly mixing processes by a fractional integral or derivative transformation. Some related remarks are made about random solutions of the Burgers equation.
The exponential rate of convergence of the distribution of the maximum of a random walk. Part II
Authors: Veraverbeke N. , Teugels J. L.
Published: July 2016

Let Gn
(x) be the distribution of the nth successive maximum of a random walk on the real line. Under conditions typical for complete exponential convergence, the decay of Gn
(x) – limn→∞ Gn
(x) is asymptotically equal to H(x) γn n–3/2
as n → ∞where γ < 1 and H(x) a function solely depending on x. For the case of drift to + ∞, G
∞(x) = 0 and the result is new; for drift to – ∞we give a new proof, simplifying and correcting an earlier version in [9].
On a quickest detection problem with variable monitoring
Authors: Balmer D. W.
Published: July 2016

The problem of detecting the arrival of a ‘disorder' in a process observed through a monitoring facility which may operate in ‘slow’ or ‘fast’ mode, is formulated as an optimal stopping problem. It is shown that in all circumstances where there may exist an optimal policy specifying the mode of observation and the time of stopping there is a unique policy satisfying certain necessary conditions of optimality; the various circumstances and control policies are specified.
Gaussian approximations to service problems: a communication system example
Authors: Gaver Donald P. , Lehoczky John P.
Published: July 2016

Messages arrive at a group of service channels in accordance with a time-dependent Poisson process. An arrival either (i) immediately begins k -stage Markovian service if an empty channel is reached, or (ii) balks and enters a retrial population if the channel sought is busy. Diffusion approximations to the number of messages in service (each stage) and in the retrial population are derived by writing stochastic differential equations (I + 0). Steady-state distributions are found and compared with certain simulation results.
On a general random exchange model
Authors: Helland Inge S. , Nilsen Trygve S.
Published: July 2016

Two independent i.i.d. sequences of random variables {Un
} and {Dn
} generate a Markov process {Xn
} by Xn
= max(Xn

–1 – Dn, Un
), n = 1, 2, …. ‘Exchange’ is defined as the event [Un > Xn

–1 – Dn
]. Conditions for existence of a limiting distribution for {Xn
} are established, and normalization is discussed when no limiting distribution exists. Finally the process {Xn
at the k th exchange; k = l, 2, …} and the time between consecutive exchanges are considered.
Computing optimal quality control policies — two actions
Authors: Wang Robert C.
Published: July 2016

We study a Markov quality control model in which we may either reset the machine or keep the machine producing items. Both discounted and average costs are considered. We shall show that a monotone policy is optimal and show how to compute the optimal critical point. Two special models are studied along with some numerical results.
Reconstruction of a stationary Gaussian process from its sign-changes
Authors: de Maré Jacques
Published: July 2016

The sign-changes over the entire real line of a Gaussian process with unit variance and mean zero are observed. The Gaussian process is reconstructed by an interpolator which is optimal in the class of linear interpolators based on that process which is one when the original process is positive and minus one when it is negative. Sufficient conditions for the interpolator to be a convolution of the sign process and a square integrable function are given. Analytical expressions of the interpolation error are derived, and the behaviour of the interpolator is studied by means of simulations.
The supersession of one rumour by another
Authors: Osei G. K. , Thompson J. W.
Published: July 2016

A model is considered for a situation in which one rumour suppresses another in a closed population. The distribution of the maximum value attained by the proportion spreading the weaker rumour is obtained in the asymptotic case, and this is compared with some actual distributions for finite population size. Closer approximations to the latter distributions are obtained.
Clustering on coloured lattices
Authors: Strauss David J.
Published: July 2016

This paper is concerned with nearest-neighbour systems on the coloured lattice (unordered state space). It extends the paper of Strauss (1975) on clustering in the two-colour case. Comparison is made with the more general methods of Besag (1974). Some tests are developed, and illustrated with an example.
Asymptotic properties of non-discrete duels
Authors: Kimeldorf George , Lang John Patrick
Published: July 2016

For fixed accuracy functions, letGnmbe a non-discrete duel in which the amounts of ammunition available to the two players arenandm, respectively. Letvnmbe the value ofGnmand let the support of the optimal pure strategies for the players be (anm, 1). The asymptotic relationships among the quantitiesn, m,vnmandanmare investigated.
Optimality of the one step look-ahead stopping times
Authors: Abdel-Hameed M.
Published: July 2016

The optimality of the one step look-ahead stopping rule is shown to hold under conditions different from those discussed by Chow, Robbins and Seigmund [5]. These results are corollaries of the following theorem: Let {Xn
, n = 0, 1, …}; X
0 = x be a discrete-time homogeneous Markov process with state space (E, ℬ). For any ℬ-measurable function g and α in (0, 1], define Aαg(x) = αExg(X
1) – g(x) to be the infinitesimal generator of g. If τ is any stopping time satisfying the conditions: Ex
[αNg(XN
)I(τ > N)]→0 as as N → ∞, then Applications of the results are considered.
An example of phase transition in countable one-dimensional Markov random fields
Authors: Cox Ted
Published: July 2016

Let S be a countable set, Q a strictly positive matrix on S × S. The set 𝒢(Q) of one-dimensional Markov random fields taking values in S with conditional probabilities determined by Q has been investigated by Spitzer [4], Föllmer [1] and Kesten [3]. In this paper a new result of Spitzer's is stated and proved, and used to present a specific example (the only one known) of a matrix Q which exhibits phase transition and admits a complete description of 𝒢 (Q).
Markovian manpower models in continuous time
Authors: Mehlmann Alexander
Published: July 2016

The problem of determining the asymptotic form of the stock vector
n
(t) in a continuous time Markovian manpower model is solved for asymptotically exponential recruitment functions {R(t)}. A new approach to the limiting behaviour of some manpower systems with given total sizes {N(t)} is then given by means of time-inhomogeneous Markov processes.
On some properties of the scan statistic on the circle and the line
Authors: Cressie Noel
Published: July 2016

The scan statistic is defined as the supremum of a particular continuous-time stochastic process, and is used as a test statistic for testing uniformity against a simple clustering type of alternative. Its distribution under the null hypothesis is investigated and weak convergence of the stochastic process to the appropriate Gaussian process is proved. An interesting link is forged between the circular scan statistic and Kuiper's statistic, which rids us of the trouble of estimating a nuisance parameter. Distributions under the alternative are then derived, and asymptotic power comparisons are made.
On the distribution of a random triangle
Authors: Alagar V. S.
Published: July 2016

In this paper we find the distribution of the area X of a randomly chosen triangle within a given triangle T. An extension of the classical Crofton technique is used in setting up a sequence of differential equations to obtain the desired distribution.
Max-infinite divisibility
Authors: Balkema A. A. , Resnick S. I.
Published: July 2016

Necessary and sufficient conditions are given for a distribution function in ℝ2 to be max-infinitely divisible. The d.f. F is max i.d. if F
t is a d.f. for every t > 0. This property is essential in defining multivariate extremal processes and arises in an approach to the study of the range of an i.i.d. sample.
Optimal replacement policy with unobservable states
Authors: Wang Robert C.
Published: July 2016

In this paper we shall solve the optimal policy for the Markovian replacement model in which the state of a machine is not observable. We shall consider both discounted and average costs and discuss two examples.
Representation of disease etiologies by certain stochastic models
Authors: Katz Leo
Published: July 2016

This paper considers the appropriateness of certain stochastic models for the representation of disease etiologies. Models for disease as the result of multiple events, and as the endpoint of a network of events, are discussed.
The advancing wave in a spatial birth process
Authors: Daniels H. E.
Published: July 2016

It is known that, in the deterministic model of a one-dimensional spatial linear birth process, a finite initial distribution of individuals develops into an advancing wave with the minimum velocity possible for such waves. In this paper, the spatial covariance density of the process is studied in the vicinity of a point moving outwards with a given velocity. Its behaviour casts doubt on the applicability of the deterministic result to individual realisations of the process.