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.

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.

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.

Consider a finite urn scheme with balls uniformly distributed amongst the urns. Using a technique of Karlin, the asymptotic behavior of the number of throws necessary to obtain at least m (m ≧ 1) balls in a given fraction of the urns is studied.

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.

Two different ways of trimming the sample path of a stochastic process in 𝔻[0, 1]: global ('trim as you go') trimming and record time ('lookback') trimming are analysed to find conditions for the corresponding operators to be continuous with respect to the (strong) J1-topology. A key condition is that there should be no ties among the largest ordered jumps of the limit process. As an application of the theory, via the continuous mapping theorem, we prove limit theorems for trimmed Lévy processes, using the functional convergence of the underlying process to a stable process. The results are applied to a reinsurance ruin time problem.

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.

The steady-state analysis of a quasi-birth-death process is possible by matrix geometric procedures in which the root to a quadratic matrix equation is found. A recent method that can be used for analyzing quasi-birth–death processes involves expanding the state space and using a linear matrix equation instead of the quadratic form. One of the difficulties of using the linear matrix equation approach regards the boundary conditions and obtaining the norming equation. In this paper, we present a method for calculating the boundary values and use the operator-machine interference problem as a vehicle to compare the two approaches for solving quasi-birth-death processes.

A model of cooperation versus defection in a sequence of games is analysed under the assumptions that the rules of the game are randomly changed from one encounter to another, that the decisions are to be made each time anew, according to the (random) rules of the specific local game, and that the result of one such game affects the ability of a player to participate and thus, cooperate in the next game. Under plausible assumptions, it is shown that all Nash solutions of the supergame determine cooperation over a non-degenerate range of rules, determining encounters of the prisoner's dilemma type.

Let {Zφ (t)} be a multitype finite-mean supercritical (CMJ) process with general characteristic φ and malthusian parameter α. It is shown that there exist some constants {c(t} such that {c–1(t)Zφ (t)} converges almost surely to a non-degenerate limit Wφ as t →∞. Characterizations of {c(t)} and Wφ are also given.

In this paper, transient characteristics related to excursions of the occupation process of M/M/∞ queues are studied, when the excursion level is large and close to the mean offered load. We show that the classical diffusion approximation by an Ornstein–Uhlenbeck (OU) process captures well the average values of the transient variables considered, while the asymptotic distributions of these variables depart from those corresponding to the OU process. They exhibit, however, equivalent tail behaviour at infinity and numerical evidence shows that they are amazingly close to each other over the whole half-line.

This paper provides Berry–Esseen rate of limit theorem concerning the number of customers in a GI/G/K queueing system observed at arrival epochs for traffic intensity ρ > 1. The main method employed involves establishing several equalities about waiting time and queue length.

We study numerical integration based on Markov chains. Our focus is on establishing error bounds uniformly on classes of integrands. Since in general state space the concept of uniform ergodicity is too restrictive to cover important cases, we analyze the error of V-uniformly ergodic Markov chains. We place emphasis on the interplay between ergodicity properties of the transition kernel, the initial distributions and the classes of integrands. Our analysis is based on arguments from interpolation theory.

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 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.

In a branching process with random environments, the probability of ultimate extinction is a function of the environment sequence, and is therefore a random variable. Explicit results about the distribution of this random variable are difficult to obtain in general. Here we assume independent and identically distributed environments and use the special properties of fractional linear generating functions to derive some explicit distributions, which may be singular or absolutely continuous, depending on the values of certain parameters. We also consider briefly tail behaviour close to 1, and provide an extension to cases where probability generating functions are not fractional linear.

We study the optimal liquidation strategy for a call spread in the case when an investor, who does not hedge, believes in a volatility that differs from the implied volatility. The liquidation problem is formulated as an optimal stopping problem, which we solve explicitly. We also provide a sensitivity analysis with respect to the model parameters.

In this paper we establish the central limit theorem for a class of stochastic partial differential equations and as an application derive this theorem for two widely studied population models: super-Brownian motion and the Fleming-Viot process.