In this paper we study a generalized Pólya urn with balls of two colors and a random triangular replacement matrix. We extend some results of Janson (2004), (2005) to the case where the largest eigenvalue of the mean of the replacement matrix is not in the dominant class. Using some useful martingales and the embedding method introduced in Athreya and Karlin (1968), we describe the asymptotic composition of the urn after the nth draw, for large n.

In this paper, we introduce a compound random mapping model which can be viewed as a generalization of the basic random mapping model considered by Ross and by Jaworski. We investigate a particular example, the Poisson compound random mapping, and compare results for this model with results known for the well-studied uniform random mapping model. We show that, although the structure of the components of the random digraph associated with a Poisson compound mapping differs from the structure of the components of the random digraph associated with the uniform model, the limiting distribution of the normalized order statistics for the sizes of the components is the same as in the uniform case, i.e. the limiting distribution is the Poisson-Dirichlet (½) distribution on the simplex {{xi} : ∑ xi ≤ 1, xi ≥ xi+1 ≥ 0 for every i ≥ 1}.

In this paper the integrated three-valued telegraph process is examined. In particular, the third-order equations governing the distributions , (where N(t) denotes the number of changes of the telegraph process up to time t) are derived and recurrence relationships for them are obtained by solving suitable initial-value problems. These recurrence formulas are related to the Fourier transform of the conditional distributions and are used to obtain explicit results for small values of k. The conditional mean values (where V(0) denotes the initial velocity of motions) are obtained and discussed.

Gene conversion is a genetic mechanism by which one gene is ‘copied and pasted’ onto another gene, where the direction can be biased between the different types. In this paper, a stochastic model for biased gene conversion within a d-unlinked multigene family and its diffusion approximation are developed for a finite Moran population. A connection with a d-island model is made. A formula for the fixation probability in the absence of mutation is given. A two-timescale argument is applied in the case of the strong conversion limit. The dual process is generally shown to be a biased voter model, which generates an ancestral bias graph for a given sample. An importance sampling algorithm for computing the likelihood of the sample is deduced.

Consider a sequence of conditionally independent Bernoulli random variables taking on the values 1 and − 1. The objective is to stop the sequence in order to maximize the discounted sum. Suppose the Bernoulli parameter has a beta distribution with integral parameters. It is optimal to stop when the conditional expectation of the next random variable is negative provided the discount factor is less than or equal to . Moreover, is best possible. The case where the parameters of the beta distribution are arbitrary positive numbers is also treated.

This paper gives an elementary proof that, under some general assumptions, the number of parts a convex set in Rd is divided into by a set of independent identically distributed hyperplanes is asymptotically normally distributed. An example is given where the distribution of hyperplanes is ‘too singular' to satisfy the assumptions, and where a different limiting distribution appears.

Queueing systems are studied with a last-come, first-served queueing discipline and batch arrivals generated by a finite number of non-exponential sources. A closed-form expression is derived for the steady-state queue length distribution. This expression has a scaled geometric form and is insensitive to the input distribution. Moreover, an algorithm for the recursive computation of the normalizing constant and the busy source distribution is presented. The results are of both practical and theoretical interest as an extension of the standard Poisson batch input case.

In economics, cash management problems may be modelled by birth-death processes which reset to central states when a boundary is reached. The nature of the transient behaviour of the probability distribution of such processes symmetric about a central state is investigated. A diffusion approximation of such processes is given and the transient probability behaviour derived from the diffusion equation.

A recursion formula is obtained by rearranging Lawrance's (1973) result concerning the superposition of independent stationary point processes for which there exist joint probability density functions for the intervals between successive points. When these component point processes are identically distributed, the formula can in principle be inverted to describe their probability structure given that of the superposition process.

We consider the Klimov model for an open network of two types of jobs. Jobs of type i arrive at station i, have processing times that are exponentially distributed with parameter µi, and when processed either go on to station j with probability pij, or depart the network with probability pi0. Costs are charged at a rate that depends on the number of jobs of the two types in the system. It is shown that for arbitrary arrival processes the policy that gives priority to those jobs for whom the rate of change of the cost function is greatest minimizes the expected cost rate at every time t. This result is stronger than the Klimov result in two ways: arrival processes are arbitrary, and the minimization is at each time t. But the result holds for only two types.

We obtain a representation of an arbitrary discrete distribution with n mass points by an equiprobable mixture of r distributions, each of which has no more than a (≧2) mass points, where r is the smallest integer greater than or equal to (n – 1)/(a – 1). An application to the generation of discrete random variables on a computer is described, which has as an important special case Walker's (1977) alias method.

A target is assumed to move according to a Wiener process in ℝ1. The probability of detecting the target is computed in terms of the search effort which accumulates along the target's path. Under certain independence assumptions this probability is given by the expectation of an exponential functional of the process. It is shown in this note that the failure probability in a search for a Wiener target is asymptotically proportional to , where T is the accumulated time spent searching. The asymptotic failure probability is also shown to be independent of the position of the search in ℝ1. In a similar fashion, it is shown that the failure probability in a search for a Wiener target in ℝ2 is independent of the position of the search and asymptotically proportional to (c log T + l)–1, c >0

This paper is concerned with the problem of first emptiness in a continuous time dam model formulated by Gani and Prabhu (1959) based on Moran's (1954) discrete time dam model. Briefly the dam model is as follows: The dam is of finite capacity K, whose content 0 ≦ Z(t) ≦ K is defined in continuous time t (0 ≦ t < ∞) by the equation where ηδt is the time the dam is empty in (t, t + δt). X(t) represents the input into the dam during time t, a Poisson process with parameter λ, such that in a small interval of time δt, the quantity δX(t) = 0 or h (< K) may be added to the dam content; min{Z(t) + δX(t),K} indicates that there will be an overflow whenever Z(t) + δX(t) > K, leaving only the amount K in the dam, and (1-η)δt represents a continuous release occurring at a steady unit rate except when z(t) = 0, when there is no release.

In a multitype critical age dependent branching process with immigration, the numbers of cell types born by t, divided by t2, tends in law to a one-dimensional (degenerate) law whose Laplace transform is explicitily given. The method of proof makes a correspondence between the moments in the m-dimensional case and the one-dimensional case, for which the corresponding limit theorem is known. Other applications are given, a possible relaxation of moment assumptions, and extensions are indicated.

Based on infinite and finite line of descent of particles of a one-dimensional supercritical Galton-Watson branching process (GWBP), we construct an associated bivariate process. We show that this bivariate process is a two-dimensional, supercritical GWBP. We also show that this process retains its branching property on appropriate probability spaces, when conditioned on set of non-extinction and set of extinction. Some asymptotic and weak convergence results for this process have been established. A generalisation of these results to a multitype p-dimensional GWBP has also been carried out.

The paper characterizes matrices which have a given system of vectors orthogonal with respect to a given probability distribution as its right eigenvectors. Results of Hoare and Rahman are unified in this context, then all matrices with a given orthogonal polynomial system as right eigenvectors under the constraint a0j = 0 for j ≥ 2 are specified. The only stochastic matrices P = {pij} satisfying p00 + p01 = 1 with the Hahn polynomials as right eigenvectors have the form of the Moran mutation model.

An M/G/1 service system with finite waiting room is studied. A customer is served by one server in phases, during some of which a place in the waiting room is occupied. The busy period length distribution is obtained from a system of integral equations leading to a linear system in Laplace-Stieltjes transforms. An asymptotic expression, for large intensity of arrival, for the expectation of this length is given. An efficiency measure giving the long run customer loss ratio is obtained. The model is shown to apply to an inventory and a container traffic problem.

This paper introduces a new form of local balance and the corresponding product-form results. It is shown that these new product-form results allow capacity constraints at the stations of a queueing network without reversibility assumptions and without special blocking protocols. In particular, exact product-form results for heavily loaded queueing networks are obtained.

We present a uniqueness result for Gibbs point processes with interactions that come from a non-negative pair potential; in particular, we provide an explicit uniqueness region in terms of activity z and inverse temperature $\beta$ . The technique used relies on applying to the continuous setting the classical Dobrushin criterion. We also present a comparison to the two other uniqueness methods of cluster expansion and disagreement percolation, which can also be applied for this type of interaction.

Consider a family of Boolean models, indexed by integers n≥1. The nth model features a Poisson point process in ℝn of intensity e{nρn}, and balls of independent and identically distributed radii distributed like X̅n√n. Assume that ρn→ρ as n→∞, and that X̅n satisfies a large deviations principle. We show that there then exist the three deterministic thresholds τd, the degree threshold, τp, the percolation probability threshold, and τv, the volume fraction threshold, such that, asymptotically as n tends to ∞, we have the following features. (i) For ρ<τd, almost every point is isolated, namely its ball intersects no other ball; (ii) for τd<ρ<τp, the mean number of balls intersected by a typical ball converges to ∞ and nevertheless there is no percolation; (iii) for τp<ρ<τv, the volume fraction is 0 and nevertheless percolation occurs; (iv) for τd<ρ<τv, the mean number of balls intersected by a typical ball converges to ∞ and nevertheless the volume fraction is 0; (v) for ρ>τv, the whole space is covered. The analysis of this asymptotic regime is motivated by problems in information theory, but it could be of independent interest in stochastic geometry. The relations between these three thresholds and the Shannon‒Poltyrev threshold are discussed.