A random graph is a collection of n points and n directed arcs: a directed arc goes equiprobably from each point to one of (n – 1) other points. m points are initially ‘infected'. We consider several schemes of epidemic process, e.g. when the infection is delivered according to arc direction. We prove that the probability of infecting all the n points with m = 1 is ∼ e/n, when n → ∞; another result is that m = o(√ n) cannot infect an essential part of the graph (having the size of O(n)). Possible applications of the models to real world phenomena are briefly discussed.

Consider a diploid population of N individuals undergoing random mating and mutation as in the infinite-allele Wright model. Choose a particular gene in generation Q for large Q, and let ft (0 ≦ t ≦ Q) be the frequency of the allelic type of the predecessor of that particular gene (or the gene itself) in generation t. By considering a ‘diffusion approximation' {xt} of {ft}, we find the distribution of the age of an allelic type now known to have frequency p, and of its distribution of frequencies since the allele came into existence. A novelty here is that the process {xt} is not a diffusion, but a process with jumps; it has x = 0 as an inaccessible entrance boundary but periodically jumps to it from the interior of [0, 1]. The formulas obtained are the same as those derived by Maruyama and Kimura, who used a totally different approach.

In this paper we generalise the so-called Yaglom conditional limit theorems to the general branching process counted by the values of a random characteristic, as suggested by Jagers (1974). Even when restricted to the special case of the usual population-size process, our results are stronger than those previously available.

A two-dimensional Poisson cluster point process is formulated by the use of a probability generating functional. Moment measures of both the cluster centre and member processes are discussed. An example is provided, and the magnitude frequency law is proved in this case.

If events occur in time according to a stochastic process then, under not very restrictive conditions, each realization will appear to come from a Poisson process with its own rate provided that the events in the realization occur at effectively random times. This result is related to de Finetti's theorem on exchangeable events. Particular applications are to the Pólya process describing accidents and the pure birth process.

The points of a homogeneous Poisson process within a compact convex set are observed. We consider how to reconstruct this domain from the observations.

A limiting distribution for the age of a class of Markov processes is found if the present state of the process is known. We use this distribution to find the age of branching processes. Using the fact that the moments of the age of birth and death processes and of diffusion processes satisfy difference equations and differential equations respectively, we find simple formulas for these moments. For the Wright–Fisher genetic model we find the probability that a given allele is the oldest in the population if all the gene frequencies are known. The proofs of the main results are based on methods from renewal theory.

A sequence of finite point processes {Pn} is constructed in using a Markov sequence of points. Essentially, in the process Pn consisting of n events, the coordinates of these events are simply the first n points of a Markov sequence suitably scaled so that the average density of the process is independent of n. The second-order properties of Pn are discussed and sufficient conditions are found for Pn to converge in distribution to a Poisson process as n →∞. A simple example involving the cardioid distribution is described.

The paper examines the renewal function associated with a sequence of probability distributions, which is defined by concave recurrence relations or by an even more general procedure. The elementary renewal theorem is generalized to such sequences. The results can be used to establish renewal theorems for first death in branching processes, if only the possibly generation dependent probability generating functions converge to a limit.

We consider maximum likelihood estimators for parameters of diffusion processes that are generated by nth-order Ito equations. We establish asymptotic consistency as well as convergence in distribution to normality for the estimators. Examples are presented and discussed.

Several new limit theorems for one- and two-sample statistics of the Kolmogorov-Smirnov type are obtained. These generalize results of Takács, Vincze, and others. The proofs combine combinatorial and weak convergence techniques.

We consider the variation in size of raw silk in the fixed size raw silk reeling process, and the relations between occurrences of feeding ends and some properties of cocoons from the stochastic viewpoint. The results obtained by stochastic analysis appear reasonable for the reeling experiment using three groups of cocoons.

The D.A.I. theorem of Gittins and Jones has proved a powerful tool in solving sequential statistical problems. A generalisation of this theorem is presented. This generalisation enables us to solve certain stochastic scheduling problems where the items or jobs to be scheduled have random times to completion, the random times having distributions dependent upon parameters to which prior distributions are allocated. Such problems are of interest in many areas where scheduling is important.

Consider a single-server queue with service times distributed as a general random variable S and with non-stationary Poisson input. It is assumed that the arrival rate function λ (·) is periodic with average value λ and that ρ = λE(S) < 1. Both weak and strong limit theorems are proved for the waiting-time process W = {W1, W2, · ··} and the server load (or virtual waiting-time process) Z = {Z(t), t ≧ 0}. The asymptotic distributions associated with Z and W are shown to be related in various ways. In particular, we extend to the case of periodic Poisson input a well-known formula (due to Takács) relating the limiting virtual and actual waiting-time distributions of a GI/G/1 queue.

Several authors have noted that simple models for the evolution of a reproducing and spatially distributed population have no limiting distribution, although a Poisson process in statistical equilibrium has sometimes been implicitly assumed. It is shown that, even when a mechanism for restricting population density is postulated, a Poisson process is usually impossible to achieve, essentially because of an assumption of independent displacements. When this assumption is abandoned, a Poisson process is possible, at least for some highly idealised models.

Black and white cells are positioned at the vertices of a rectangular lattice. When a cell division occurs, the daughter cells are of the same colour as the parent cell; one of them replaces an adjacent cell and the other remains in the position of the parent cell. In one variant of the model it is assumed that whenever a white cell appears at the origin it is transformed into a black cell; apart from this the black and white cells are equally competitive and in particular they divide at the same rate. Initially, only the cell at the origin is black. The asymptotic behaviour of the black clone is investigated.

A population is composed of an infinite number of colonies situated at the integer points of a single co-ordinate axis. Each colony develops according to a simple birth and death process and migration is allowed between nearest neighbours only. Exact expressions are derived for both the asymptotic velocity of propagation and the shape of the wave profile as the population spreads along the axis, and it is shown that they are identical with the saddle point approximations developed by Daniels.

An explicit formula is derived for the variance normalizing constant in the central limit theorem for chain-dependent processes. As an application to meteorology, a specific chain-dependent process is proposed as a probabilistic model for the sequence of daily amounts of precipitation. This model is a generalization of the commonly used Markov chain model for the occurrence of precipitation.

For a battle governed by the stochastic version of Lanchester's square law the transient distribution of state is found; also various terminal distributions. The terminal distributions arise also in certain related random walks.

Using the expectation recurrence equation of discrete-time Markovian manpower models, an asymptotic relation for the population numbers in various grades is derived.