We obtain results connecting the distribution of the random variables Y and W in the supercritical generalized branching processes introduced by Crump and Mode. For example, if β > 1, EYβ and EWβ converge or diverge together and regular variation of the tail of one of Y, W with non-integer exponent β > 1 is equivalent to regular variation of the other. We also prove analogous results for the continuous-time continuous state-space branching processes introduced by Jirina.

The model is just the natural extension to n dimensions of the classical model for an epidemic with removals: however, the deterministic version does show a richer variety of threshold behaviour. In the stochastic model, some results will be discussed for the embedded Markov chain, and also a possible control policy involving the removal of entire communities.

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.

The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for ‘completely’ regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the ‘fermion’ process that is suitable whenever the points are repulsive.

The techniques used by Doeblin and Chung to obtain ordinary limit laws (central limit laws, weak and strong laws of large numbers, and laws of the iterated logarithm) for Markov chains, are extended to obtain analogous functional limit laws for stochastic processes which have embedded processes satisfying these laws. More generally, it is shown how functional limit laws of a stochastic process are related to those of a process embedded in it. The results herein unify and extend many existing limit laws for Markov, semi-Markov, queueing, regenerative, semi-stationary, and subordinated processes.

Variability in space of plankton can arise from stochastic effects on growth and predation, and should be smoothed by horizontal diffusion of the water. The interactions of these factors within a non-linear system should permit small-scale perturbations to appear as larger-scale patches. A simple simulation illustrates this possibility.

Over the last four years the Forestry Commission has conducted sample surveys to estimate the spread of Dutch Elm disease in England and Wales. The data were described and some of the problems of making inferences from them were indicated. These include the definition of an individual elm tree, the two main modes of infection, the complexity of the spatial distribution of elms and the existence of at least two strains of the fungus C. ulmi with different pathogenicity.

Multidimensional sampling of real data, e.g., in space and time, often requires observations at non-uniformly spaced intervals. A discrete transform of a multidimensional stationary stochastic process transforms a multivariate problem into an asymptotically univariate one if the spacing is uniform in at least one dimension. For both uniform and non-uniform sampling and a model of ‘signal’ imbedded in a ‘noise’ process, asymptotic normality and independence justifies statistical testing in each cell of the transformed domain of the hypothesis ‘noise alone’ versus the alternate ‘signal plus noise’.

In this paper, the authors consider a class of stochastic systems described by Ito differential equations for which both controls and parameters are to be chosen optimally with respect to a certain performance index over a fixed time interval. The controls to be optimized depend only on partially observed current states as in a work of Fleming. However, he considered, instead, a problem of optimal control of systems governed by stochastic Ito differential equations with Markov terminal time. The fixed time problems usually give rise to the Cauchy problems (unbounded domain) whereas the Markov time problems give rise to the first boundary value problems (bounded domain). This fact makes the former problems relatively more involved than the latter. For the latter problems, Fleming has reported a necessary condition for optimality and an existence theorem of optimal controls. In this paper, a necessary condition for optimality for both controls and parameters combined together is presented for the former problems.

Competition models for spatially distributed individuals may be developed from the basic assumption that each individual will exert a competitive influence within a circle, centered on the individual's location and with radius determined by a random variable, R (Gerrard (1969)).

A criterion for ergodicity of lattice interactions has been given by Dobrushin [2], and improved by Harris [3] and Holley [5]. In this paper we present a simplified derivation of these results, and obtain stronger conditions for interactions on the one- and two-dimensional integer lattices, and the 3–Bethe lattice.

Some years ago Hammersley and Clifford considered the problem of assigning probabilities to colourings of a graph according to certain local conditions. Subsequent expositions such as those of Grimmett, Besag and Preston, whilst providing more elegant derivations, have shown that the assignment of probabilities coincides with the limiting distribution of spatial-temporal processes arising from theoretical physics. There has been relatively little work on spatial-temporal processes outside this class. In this paper the approximate behaviour of an alternative class of processes is investigated. Topics will include statistical estimation of parameters and existence of unique equilibria.

For many years animal behaviourists, sociologists and the like have been studying how animals and humans form groups, and how the groups change in time. A brief review was given of some of the models devised for such group changing situations, and some new models were described.