A general time-dependent stochastic compartmental model is considered, where particles may move between a set of m compartments or out of the system, and new particles may be introduced into the compartments by immigration. A simple argument not relying on generating function techniques is given for the solution of such a system. It is then demonstrated that this class of compartmental models can be used as approximations to more complex stochastic systems by replacing dependence on certain stochastic variables by dependence on corresponding deterministic variables, with some recent examples being discussed. A compartmental approximation to the general stochastic epidemic is then constructed which appears on the basis of some simulations to compare favourably with the true process, particularly after the infection has got established in the population.

Leslie (1945) models the evolution in discrete time of a closed, single-sex population with discrete age groups by multiplying a vector describing the age structure by a matrix containing the birth and death rates. We suppose that successive matrices are chosen according to a Markov chain from a finite set of matrices. We find exactly the long-run rate of growth and expected age structure. We give two approximations to the variance in age structure and total population size. A numerical example illustrates the ergodic features of the model using Monte Carlo simulation, finds the invariant distribution of age structure from a linear integral equation, and calculates the moments derived here.

In Pickard (1976) limit theorems were obtained for the classical Ising model at non-critical points. These determined the asymptotic distribution of the sample nearest-neighbour correlation, thereby providing a basis for statistical inference by confidence intervals. In this paper, these limit theorems are extended to the statistically significant case of different vertical and horizontal interactions. Results at critical points are also obtained. Critical points clearly have the potential to seriously distort statistical inferences, especially in their immediate neighbourhoods. For our Ising model it turns out that such distortion is relatively minor. Surprisingly, in the two-parameter case the correlation between the sufficient statistics exhibits peculiar asymptotic behaviour resulting in a singular covariance matrix at critical points in the central limit theorem. Finally, at critical points, unusual norming constants are required for the central limit theorem, and our results are much more sensitive to the relative rate at which m, n tend to infinity.

The purpose of this paper is to give a short introduction to the theory of point processes and random measures. Our hope is that the paper can be used as a complement to a study of Billingsley's book Convergence of Probability Measures. The subjects treated are: topological properties of the space of realizations; characterization and convergence of random measures with special attention to simple point processses and diffuse random measures; the relation between the topology used and the Skorohod topology. Some thinning and superposition results are given without proofs.

In two earlier papers [6], [7] the properties of bivariate renewal processes and their associated two-dimensional renewal functions, H(x, y) were examined. By utilising the Fréchet bounds for joint distributions and the properties of univariate renewal processes, a collection of upper and lower bounds for H(x, y) are constructed. The evaluation of these bounds is carried out for the case of the family of bivariate Poisson processes. An interesting by-product of this investigation leads to a new inequality for the median of a Poisson random variable.

The paper deals with asymptotic properties of the transition probabilities of a countable non-homogeneous Markov chain, the main concept used in the proofs being that of the tail σ-field of the chain. A state classification similar to that existing in the homogeneous case is given and a strong ratio limit property is shown to parallel the basic limit theorem for positive homogeneous chains. Some global asymptotic properties for null chains are also derived.

A simple expression for the characteristic functional of generalized shot noise is developed. Through expansions in terms of functional derivatives this yields expressions for moment functions of all orders. A central limit theorem also follows. Several examples are discussed.