The classical two-player gambler's ruin problem is well known and has attracted a great deal of attention in the literature in various forms. Extensions of these two-player problems to three or more players do not appear to have been discussed however, and this paper is concerned with one particular extension of the problem to the case of three players.

We introduce a general two-colour interacting urn model on a finite directed graph, where each urn at a node reinforces all the urns in its out-neighbours according to a fixed, non-negative, and balanced reinforcement matrix. We show that the fraction of balls of either colour converges almost surely to a deterministic limit if either the reinforcement is not of Pólya type or the graph is such that every vertex with non-zero in-degree can be reached from some vertex with zero in-degree. We also obtain joint central limit theorems with appropriate scalings. Furthermore, in the remaining case when there are no vertices with zero in-degree and the reinforcement is of Pólya type, we restrict our analysis to a regular graph and show that the fraction of balls of either colour converges almost surely to a finite random limit, which is the same across all the urns.

We consider a preferential duplication model for growing random graphs, extending previous models of duplication graphs by selecting the vertex to be duplicated with probability proportional to its degree. We show that a special case of this model can be analysed using the same stochastic approximation as for vertex-reinforced random walks, and show that ‘trapping’ behaviour can occur, such that the descendants of a particular group of initial vertices come to dominate the graph.

We review some urn and random-allocation models, mostly using probability generating function (PGF) methods. We begin by formulating a basic problem which can be thought of as either an urn or a random-allocation model; a PGF solution to it is outlined. When the compartments in the latter model are no longer homogeneous, the multivariate PGF can still be derived, though the algebra becomes cumbersome. Some results are given for the case where there are two types of compartment and for the case where there are two types of ball. Some comments are offered on the Frobenius–Harper property of PGFs.

The evolution of strategies in animal contests is examined where the dynamical equation takes into account population growth rates. This leads to a different definition of evolutionary stable strategy (ESS) from the usual one. Consequences for independent haploid species are then contrasted with the previous theory. Inheritance patterns for male–female contests with sex-dependent payoffs are considered. In particular, if males and females evolve independently to the same ESS, then so does the diploid species under random mating. Finally, the evolution of diploid populations where strategies are determined at a diallelic locus is investigated.

Consider N sources, each of which transmits signals according to a renewal process with a common, known, underlying distribution. Sequential point estimators for N, which are asymptotically efficient in comparison to the best fixed stopping strategy, are derived. Confidence statements for N, and applications to the problem of estimating finite populations are discussed.

The exact limiting distribution, and moments, of the stopping time we use, which was suggested by Starr, are derived.

Consider a two-component parallel system. There is a stockpile of n spares, spare i having lifetime Xi, exponentially distributed with rate µi. The system breaks down when neither component functions. Every time a spare expires it has to be replaced, to avoid system breakdown. We investigate the policies which maximize the expected lifetime of the system using these n spares.

Transform inversions, in which density and survival functions are computed from their associated moment generating function $\mathcal{M}$, have largely been based on methods which use values of $\mathcal{M}$ in its convergence region. Prominent among such methods are saddlepoint approximations and Fourier-series inversion methods, including the fast Fourier transform. In this paper we propose inversion methods which make use of values for $\mathcal{M}$ which lie outside of its convergence region and in its analytic continuation. We focus on the simplest and perhaps richest setting for applications in which $\mathcal{M}$ is either a meromorphic function in its analytic continuation, so that all of its singularities are poles, or else the singularities are isolated essential. Asymptotic expansions of finite- and infinite-orders are developed for density and survival functions using the poles of $\mathcal{M}$ in its analytic continuation. For finite-order expansions, the expansion error is a contour integral in the analytic continuation, which we approximate using the saddlepoint method based on following the path of steepest descent. Such saddlepoint error approximations accurately determine expansion errors and, thus, provide the means for determining the order of the expansion needed to achieve some preset accuracy. They also provide an additive correction term which increases accuracy of the expansion. Further accuracy is achieved by computing the expansion errors numerically using a contour path which ultimately tracks the steepest descent direction. Important applications include Wilks’ likelihood ratio test in MANOVA, compound distributions, and the Sparre Andersen and Cramér–Lundberg ruin models.

In [4], the limiting behaviour of a stochastic system with two types of input was investigated by reducing the problem to the solution of an integral equation. In this note we use the same approach to study the equilibrium waiting time problem for the general single server queue with preemptive service interruptions. (For a comprehensive account of the existing literature on queues with service interruptions we refer to [2] and [3].)

We derive asymptotic forms for the distributions of k-point scan statistics as the interval L under study becomes infinite, while k and the window length are held fixed. In the Poisson case the intensity is also held fixed. In the uniform case the number of points N becomes infinite and N/L tends to a limit, representing a limiting intensity. These results are made explicit for k = 3, and in the Poisson case provide approximations which are typically accurate to six or seven figures, even for small L.

A generalisation of the classical general stochastic epidemic within a closed, homogeneously mixing population is considered, in which the infectious periods of infectives follow i.i.d. random variables having an arbitrary but specified distribution. The asymptotic behaviour of the total size distribution for the epidemic as the initial numbers of susceptibles and infectives tend to infinity is investigated by generalising the construction of Sellke and reducing the problem to a boundary crossing problem for sums of independent random variables.

There are now many examples in various fields where the behaviour of ‘particles' as exhibited by their transition from one state to another is described by a multidimensional stochastic process. The linear migration model in which particles move independently of one another through a number of states has been dealt with by Bartlett (1949). This process has been used by Siegert (1949) in considering the approach to equilibrium of non-interacting gas molecules and by Krieger and Gans (1960) and Gans (1960) to examine the distribution of a multicomponent system disturbed from its equilibrium distribution and relaxing by first-order processes to another equilibrium. The correspondence between the deterministic model based on the principle of mass action and the stochastic model has been discussed by Darvey and Staff (1966) in the context of unimolecular multicomponent chemical reactions.

For i.i.d. random variables in the domain of attraction of a max-stable distribution with discount and observation costs we determine asymptotic approximations of the optimal stopping values and asymptotically optimal stopping times. The results are based on Poisson approximation of related embedded planar point processes. The optimal stopping problem for the limiting Poisson point processes can be reduced to differential equations for the boundaries. In several cases we obtain numerical solutions of the differential equations. In some cases the analysis allows us to obtain explicit optimal stopping values. This approach thus leads to approximate solutions of the optimal stopping problem of discrete time sequences.

Consider an array of binary random variables distributed over an m1(n) by m2(n) rectangular lattice and let Y1(n) denote the number of pairs of variables d, units apart and both equal to 1. We show that if the binary variables are independent and identically distributed, then under certain conditions Y(n) = (Y1(n), · ··, Yr(n)) is asymptotically multivariate normal for n large and r finite. This result is extended to versions of a model which provide clustering (repulsion) alternatives to randomness and have clustering (repulsion) parameter values nearly equal to 0. Statistical applications of these results are discussed.

A queueing system is considered where each item has a property associated with it, and where the service time interposed between two items depends on the properties of both of these items. The steady state of a single-channel queue of this type, with Poisson input, is investigated. It is shown how the probability generating function of the number of items waiting can be found. Easily applied approximations are given for the mean number of items waiting and for the average waiting time.

If T is a stopping time on a martingale {St}, the mth order moments of T and ST (for integers m ≧ 2) are related by an identity known as a Wald equation of order m. Wald equations hold under certain conditions, which are simplified in this paper by showing that among the set of previously known conditions, some are implied by the others.

The kin number problem in its simplest form is that of the relationship between sibship sizes and offspring numbers. The fact that the distributions are different, and the relationship between the two, is well known to demographers. It is important in such applications as estimating fertility from sibship rather than offspring counts. Further studies have been made, concerning relatives of other degrees of affinity than siblings, but these did not usually yield joint distributions. Recently this aspect of the problem has been studied in the framework of a Galton–Watson process (Waugh (1981), Joffe and Waugh (1982)).

In these studies the population is treated as monotype. Applications such as pedigree studies of diseases require a multitype approach (in the example, two types: victims and others). In this paper such a study is undertaken. The existence of more than one type complicates the time-reversal used in the previous studies, and raises questions about the way in which the focal individual (called ‘Ego') is sampled. These are dealt with, and joint distributions obtained, under a number of sampling schemes which might arise naturally.

An error bound for convergence to the Ewens sampling formula is given where the population size or mutation rate may vary from generation to generation, or the population is not yet at equilibrium. An application is given to a model of Hartl and Campbell about selectively-equivalent subtypes within a class of deleterious alleles, and a theorem is proven showing that the size of the deleterious class stays within bounds sufficient to apply the first result. Generalizations are discussed.

We consider continuous review inventory systems with general doubly stochastic Poisson demand. In this specific case the demand rate, experienced by the system, varies as a function of the age of the oldest unit in the system. It is known that the stationary distributions of the ages in such models often have a strikingly simple form. In particular, they exhibit a typical feature of a Poisson process: given the age of the oldest unit the remaining ages are uniform. The model we treat here generalizes some known inventory models dealing with partial backorders, perishable items, and emergency replenishment. We derive the limiting joint density of the ages of the units in the system by solving partial differential equations. We also answer the question of the uniqueness of the stationary distributions which was not addressed in the related literature.