Some Wiener-Hopf type results are collected, related and given more direct proofs. Spitzer's random-walk method for the Wiener-Hopf integral equation also produces his factorisation relating functionals of maxima and minima. Transform equations are interpreted as decompositions of time-changed processes. Discrete- and continuous-time versions are related. Prabhu's factorisation for generators is equivalent to Fristedt's Lévy measure factorisation and to process decomposition.

A simple direct proof is given of Minkowski's result that the mean length of the orthogonal projection of a convex set in E3 onto an isotropic random line is (2π)–1 times the integral of mean curvature over its surface. This proof is generalised to a correspondingly direct derivation of an analogous formula for the mean projection of a convex set in En onto an isotropic random s-dimensional subspace in En. (The standard derivation of this, and a companion formula, to be found in Bonnesen and Fenchel's classic book on convex sets, is most indirect.) Finally, an alternative short inductive derivation (due to Matheron) of both formulae, by way of Steiner's formula, is presented.

A type of service system of particular interest in teletraffic theory has the property that calls (customers) originating from different sources have access to different subsets of a single group of devices (servers). One may or may not allow the formation of queues. Thierer [11], [12] has studied such a system with queues assuming it to be reversible. We investigate the conditions under which this assumption is exact.

In this paper we develop a model for describing the spread of the parasitic disease bilharziasis. The detailed model first developed (Model A) is made more tractable by replacing infecting population sizes by their expected values in the transition probabilities, giving Model B. Using this approximating model it is possible to examine the nature of the spread of the disease and the effects of control measures. The effects of a particular control measure are examined in order to compare Models A and B.

In their paper [2] Linnik and Zinger proved a theorem which has an important role in both theoretical and applied statistics. For the proof of their theorem the authors used a condition which greatly narrows down its applicability. In this paper we prove the theorem of Linnik and Zinger without this condition, and also give other extensions of it.

A sequence of random variables η0, η1, …, ηn, … is defined by the recurrence formula ηn = max (ηn–1 + ξn, 0) where η0 is a discrete random variable taking on non-negative integers only and ξ1, ξ2, … ξn, … is a semi-Markov sequence of discrete random variables taking on integers only. Define Δ as the smallest n = 1, 2, … for which ηn = 0. The random variable ηn can be interpreted as the content of a dam at time t = n(n = 0, 1, 2, …) and Δ as the time of first emptiness. This paper deals with the determination of the distributions of ηn and Δ by using the method of matrix factorisation.

We establish the estimation equations corresponding to a system in which the state is a process with jumps and the observation is a point process. We then solve two problems: the estimation of the state of a queue when its output is observed, and the estimation of the time of disorder of a machine when the ‘complaints process’ is observed.

A statistical method for estimating the factorisation of the spectral density of a stationary point process is presented and asymptotic properties of the resulting estimates are derived. The estimated functions are of interest in the analysis of a self-exciting process and more generally in the problem of linear prediction, and can be viewed as an alternative second-order analysis of the process. Some examples are discussed.

Our aim here is to give a survey of that part of continuous-time fluctuation theory which can be approached in terms of functionals of Lévy processes, our principal tools being Wiener-Hopf factorisation and local-time theory. Particular emphasis is given to one- and two-sided exit problems for spectrally negative and spectrally positive processes, and their applications to queues and dams. In addition, we give some weak-convergence theorems of heavy-traffic type, and some tail-estimates involving regular variation.

The General Stochastic Epidemic model is extended to allow for outside infection. The likelihood is derived for small households, and the difficulties for larger populations are discussed.

Aggregated spatial patterns may be generated by a clustering process (see, for example, Bartlett (1964)) in which ‘parent’ events are distributed completely at random, and produce, independently, random numbers of ‘offspring’ according to some distribution Pn; the position of each offspring relative to its parent is governed, independently, by a given bivariate distribution. Parents and offspring are assumed indistinguishable. For such a process, Bartlett (1974) shows that the distribution function F of the distance, X say, from a randomly selected point to the nearest event is given by where ρ denotes the mean number of parents per unit area, A is the circle with centre the origin and radius x, and E(ds) denotes the event ‘no offspring in A from parent in ds’.

A simple path transformation is described which connects one-dimensional Brownian motion with the radial part of three-dimensional Brownian motion. This provides simple proofs of various path decompositions for these processes described by David Williams.

The practical problems involved in the maximum likelihood estimation of the parameters in a binary Markov field on a lattice are investigated. Methods using whole and conditional data are compared.

The merits of a conditional probability (or Markovian) approach to the analysis of local spatial interaction lie in its intuitive appeal and its ability to cope with non-Gaussian situations. Whilst relationships between plausible spatial-temporal models and Markov fields may sometimes be obtained, this does not remove the need for a purely spatial approach to be available. In addition to promoting the above point of view, some Monte Carlo simulation results, due to Mr. D. H. Green, were presented. Analytical progress on coding efficiency was reported.