This paper is concerned with a model for the effect of erosion on crop production. Crop yield in the year n is given by X(n) = YnLn, where is a sequence of strictly positive i.i.d. random variables such that E{Y 1} <∞, and is a Markov chain with stationary transition probabilities, independent of . When suitably normalized, leads to a martingale which converges to 0 almost everywhere (a.e.) as n → ∞. In addition, for large n, the distribution of Ln is approximately lognormal. The conditional expectations and probabilities of , given the past history of the process, are determined. Finally, the asymptotic behaviour of the total crop yield is discussed. It is established that under certain regularity conditions Sn converges a.e. to a finite-valued random variable S whose Laplace transform can be obtained as the solution of a Volterra-type linear integral equation.

We determine the decay rate of the survival probability of subcritical branching processes in a two-state random environment, where one state is subcritical, the other supercritical. This result is applied to obtain the asymptotic behavior (as n →∞) of the number of different words of length n occurring on the binary, and generally the b-ary, tree with Bernoulli percolation.

Let Vk,n be the number of vertices of degree k in the Euclidean minimal spanning tree of Xi , , where the Xi are independent, absolutely continuous random variables with values in Rd. It is proved that n –1 Vk,n converges with probability 1 to a constant α k,d. Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of Vk,n.

Consequences of embedding sequences {Mn } in an extremal-F process are discussed where Mn represents the maximum of n independent (but not necessarily identically distributed) random variables. Various limit theorems are proved for the sample record rate, record times, inter-record times, and record values. These results are illustrated with applications to three particular record models: the Yang (1975) record model where population size increases geometrically, a record model where linear improvement is present, and a record model incorporating features of the previous two.

Maximum stability of a distribution with respect to a positive integer random variable N is defined by the property that the type of distribution is not changed when considering the maximum value of N independent observations. The logistic distribution is proved to be the only symmetric distribution which is maximum stable with respect to each member of a sequence of positive integer random variables assuming value 1 with probability tending to 1. If a distribution is maximum stable with respect to such a sequence and minimum stable with respect to another, then it must be logistic, loglogistic or ‘backward' loglogistic. The only possible sample size distributions in these cases are geometric.

The limiting behaviour of the convex hull of a sample in is studied using the support function. Results like that of Eddy and Gale (1981) are proved without the condition of spherical symmetry from that paper.

The chord length distributions of planar convex sets are discussed, particularly the density values at the extremes of the range; there is a qualitative distinction between polygons and sets with smooth boundaries. The distance between convex sets is related to the distance between distribution functions.

A saddlepoint expansion is given for conditional probabilities of the form where is an average of n independent bivariate random vectors. A more general version, corresponding to the conditioning on a p – 1-dimensional linear function of a p-dimensional variable is also included. A separate formula is given for the lattice case. The expansion is a generalization of the Lugannani and Rice (1980) formula, which reappears if and are independent. As an example an approximation to the hypergeometric distribution is derived.

We study the transition probabilities of the diffusions dXt = (1 – exp(Xt ))dt + dWt and dXt = – tanh Xtdt + dWt , in terms of special functions of mathematical physics (confluent hypergeometric and Legendre functions, respectively).

Computer scientists have introduced ‘paging algorithms' which are a special class of Markov chains on permutations known, in probability theory, as ‘libraries': books being placed on a shelf T (T is an infinite interval of the set Z of the integers) and a policy ρ : T → T such that ρ (t) < t being chosen, a book b placed at t ∊ T is selected with probability pb , it is removed and replaced at ρ (t) prior to next removal. The different arrangements of books on the shelf are the states of the Markov chain. In this paper we prove that, if the shelf is not bounded on the left, any library (i.e. for any policy ρ and any probability ρ on the books) is transient.

A correlated random walk is studied in which, at each stage, the velocity changes according to a first-order process. Motion is considered both with and without friction, the former situation being the discrete analogy of the Uhlenbeck–Ornstein process. Exact and limiting expressions are developed for the cumulant structures.

Cowan and Zabczyk (1978) have studied a continuous-time generalization of the so-called secretary problem, where options arise according to a homogeneous Poisson processes of known intensity λ. They gave the complete strategy maximizing the probability of accepting the best option under the usual no-recall condition. In this paper, the solution is extended to the case where the intensity λ is unknown, and also to the case of an inhomogeneous Poisson process with intensity function λ (t), which is either supposed to be known or known up to a multiplicative constant.

The theory of robust non-linear filtering in Clark (1978) and Davis (1980), (1982) is used to evaluate the limiting conditional distribution of a diffusion, given an observation of a ‘rare-event' sample-path of the diffusion, as the signal-to-noise ratio and the diffusion noise-intensity converge to infinity and zero respectively. Under mild conditions it is shown that the limiting conditional distribution is a Dirac measure concentrated at a trajectory which solves a variational problem parametrised by the sample-path of the observed signal.

This paper considers single-server, multi-queue systems with cyclic service. Non-zero switch-over times of the server between consecutive queues are assumed. A stochastic decomposition for the amount of work in such systems is obtained. This decomposition allows a short derivation of a ‘pseudo-conservation law' for a weighted sum of the mean waiting times at the various queues. Thus several recently proved conservation laws are generalised and explained.

We discuss the quasi-stationary distribution obtained when a simple birth and death process is conditioned on never exceeding K. An application of this model to one-dimensional circuit-switched communication networks is described, and some special cases examined.

Suppose that pulses arrive according to a Poisson process of rate λ with the duration of each pulse independently chosen from a distribution F having finite mean. Let X(t) be the shot noise process formed by the superposition of these pulses. We consider functionals H(X) of the sample path of X(t). H is said to be L-superadditive if for all functions f and g. For any distribution F for the pulse durations, we define H(F) = EH(X). We prove that if H is L-superadditive and for all convex functions ϕ, then . Various consequences of this result are explored.

A class of M/G/1 time-sharing queues with a finite number of service positions and unlimited waiting space is described. The equilibrium distribution of symmetric queues belonging to this class is invariant under arbitrary service-independent reordering of the customers at instants of arrivals and departures. The delay time distribution, in the special case of one service position where preempted customers join the end of the line, is provided in terms of Laplace transforms and generating functions. It is shown that placing preempted customers at the end of the line rather than at the beginning of the line results in a reduction of the delay time variance. Comparisons with the delay time variance of the case of unlimited number of service positions (processor sharing system) are presented.

The questions of almost sure convergence of the sequence Xn defined below in (1) as well as that of to a finite limit are settled here. These sequences arise in a simple stochastic model for crop erosion considered by Todorovic and Gani (1987).

Our concern is with a particular problem which arises in connection with a discrete-time Markov chain model for a graded manpower system. In this model, the members of an organisation are classified into distinct classes. As time passes, they move from one class to another, or to the outside world, in a random way governed by fixed transition probabilities. In this paper, the emphasis is placed on evaluating exact values of the probabilities of attaining and maintaining a structure.