The solution of a hierarchical n-compartmental homogeneous process with multiple-valued migration rates is obtained using Laplace transform techniques. Such models are suggested for modelling the infection process of endemic diseases because of their property of gamma-distributed compartmental residence times in each of n compartments. Its relevance to the onchocerciasis infection process (with and without the imposition of controls) is discussed and application to age-prevalence data (having only two infective states into which an individual can be classified) is undertaken.

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{Y1} <∞, 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.

Every realization of a Poisson line process is a set of lines which subdivides the plane into a population of non-overlapping convex polygons. To explore the unknown statistical features of this population, an alternative stochastic construction of random polygons is developed. This construction, which is based on an alternating sequence of random angles and side lengths, provides a fast simulation method for obtaining a random sample from the polygon population. For the isotropic case, this construction is used to obtain a random sample of 2500000 polygons, providing the most precise estimates to date of some of the unknown distributional characteristics.

Algorithms for a stochastic population process, based on assumptions underlying general age-dependent branching processes in discrete time with time inhomogeneous laws of evolution, are developed through the use of a new representation of basic random functions involving birth cohorts and random sums of random variables. New algorithms provide a capability for computing the mean age structure of the process as well as variances and covariances, measuring variation about means. Four exploratory population projections, testing the implications of the algorithms for the case of time-homogeneous laws of evolution, are presented. Formulas extending mean and variance functions for unit population projections to an arbitrary initial population size are also presented. These formulas show that, in population processes with non-random laws of evolution, stochastic fluctuations about the mean function are negligible when initial population size is large. Further extensions of these formulas to the case of randomized laws of evolution suggest that stochastic fluctuations about the mean function can be significant even for large initial populations.

In this paper we obtain an upper bound for the maximum of random fields of the form , where CP denotes circles of fixed radius and dW(P′) is a plane white noise field.

The results presented are obtained by means of successive steps involving Slepian's lemma for random fields, inequalities on Brownian fields and planar stochastic integrals.

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.

A Malthusian parameter for the generation-dependent general branching process is introduced and some asymptotics of the expected population size, counted by a general non-negative characteristic, are discussed. Processes both with and without immigration are considered.

In this paper a random motion on the surface of the 3-sphere whose probability law is a solution of the telegraph equation in spherical coordinates is presented. The connection of equations governing the random motion with Maxwell equations is examined together with some qualitative features of its sample paths. Finally Brownian motion on the 3-sphere is derived as the limiting process of a random walk with latitude-changing probabilities.

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–1Vk,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.

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.

A Poisson limit theorem for sums of dissociated 0–1 random variables is refined by deriving the first terms in an asymptotic expansion. The most natural refinement does not remove all the first-order error in a number of applications to tests of clustering, and a further approximation is derived which gives excellent results in practice. The proofs are based on the technique of Stein and Chen.

We consider the problem of approximating the stationary distribution of a positive-recurrent Markov chain with infinite transition matrix P, by stationary distributions computed from (n × n) stochastic matrices formed by augmenting the entries of the (n × n) northwest corner truncations of P, as n →∞.

Solutions of the Burgers equation with a stationary (spatially) stochastic initial condition are considered. A class of limit laws for the solution which correspond to a scale renormalization is considered.

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.

A generalization of a birth and death chain in a random environment (Yn, Zn) is developed allowing for feedback to the environmental process (Yn). The resulting process is then known as a birth and death chain in a random environment with feedback. Sufficient conditions are found under which the (Zn) process goes extinct almost surely or has strictly positive probability of non-extinction.

We obtain a sufficient condition for the transience of a Markov chain, and a sufficient condition for its null recurrence. These are applied to characterize the stability of a multiple-access communication system. Performance bounds for the system are also obtained.

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.

A continuum structure function γ is a non-decreasing mapping from the unit hypercube to the unit interval. Block and Savits (1984) use the sets and to determine bounds on the distribution of γ (X) when X is a vector of associated random variables. It is shown that, if γ admits of a modular decomposition, improved bounds may be obtained.

In this paper we consider an economic agent (the consumer) having initial wealth x(0), which he is allowed to spend during the time interval [0, T]. There are N consumer goods, the prices of which are described by a system of stochastic differential equations, and the consumer can spend his money as he pleases provided that he does not get into debt. The consumer's problem is to maximize his expected utility over [0, T]. Given a separable utility function we show that the optimal consumption policy is linear in the wealth variable regardless of the structure of the price system, and if we assume that the price processes are geometric Brownian motions we can give the optimal consumption policy explicitly.