The Skorokhod representation for martingales is used to obtain a functional central limit theorem (or invariance principle) for martingales. It is clear from the method of proof that this result may in fact be extended to the case of triangular arrays in which each row is a martingale sequence and the second main result is a functional central limit theorem for such arrays. These results are then used to obtain two functional central limit theorems for processes with stationary ergodic increments following on from the work of Gordin. The first of these theorems extends a result of Billingsley for Φ-mixing sequences.

A population model is constructed which combines the ideas of a discrete time branching process with random environments and a continuous time non-homogeneous Markov branching process. The extinction problem is considered and necessary and sufficient conditions for extinction are determined. Also discussed are limit theorems for what corresponds to the supercritical case.

This work studies the long-term effects of mutation and selection pressures on a diploid population embracing many genetic types. A number of results previously established for the simpler asexual case (see [4]) are extended to the cases of random mating and complete inbreeding (Theorem 1), and then, under particular conditions, to certain circumstances of mixed random mating and inbreeding (Theorem 3 and Corollary 1). Several implications for sex and diploidity are drawn from Theorem 2 and its corollaries. Further biological interpretations of these findings, especially of Theorem 2, are given in [3].

If x0 is a particular state for a continuous-time Markov process X, the random time set is often of both practical and theoretical interest. Ignoring trivial or pathological cases, there are four different types of structure which this random set can display. To some extent, it is possible to treat all four cases in a unified way, but they raise different questions and require different modes of description. The distributions of various random quantities associated with can be related to one another by simple and useful formulae.

A general discussion and survey is provided of the characterization of the normal distribution by the identical distribution of linear forms. The first result dates to 1923 when Pólya showed that if X and Y are i.i.d. random variables satisfying certain conditions, and if aX + bY is distributed as (a2 + b2)1/2X, then X has a normal distribution. This result has been generalized in several directions. In addition to a recasting of some of the results, an extension in the multivariate case is provided.

Using some well-known and some recently proved asymptotic properties of regenerative processes, we present a new proof in a general regenerative setting of the equivalence of the limiting distributions of a stochastic process at an arbitrary point in time and at the time of an event from an associated Poisson process. From the same asymptotic properties, several conservation equations are derived that hold for a wide class of GI/G/1 priority queues. Finally, focussing our attention on the alternating-priority queue with Poisson arrivals, we use both types of result to give a new, simple derivation of the expected steady-state delay in the queue in each class.

The objective of this work is to study the long range evolutionary traits in a population with an infinite number of types; we are especially interested in the asymptotic rate of evolution, variance and type-distribution. In this paper we concentrate on an asexual population with the same probabilistic mutation force acting over all types in the population.

Most results are proved under the assumption that the mutation effect is also unchanged in time; some are extended to more general cases with time-changed distribution of mutation.

A branching process with variable lifetime distribution is defined by a sequence of distribution functions {Gi(t)}, together with a probability generating function, h(s) = Σk∞= 0pksk. An ith generation particle lives a random length of time, determined by Gi(t). At the end of a particle's life it produces children, the number being determined by h(s). These offspring behave like the initial particle except they are (i + 1)th generation particles and have lifetime distribution Gi + 1 (t).

Let Zi(t) be the number of particles alive at time t, the initial particle being born into the ith generation. Integral equations are derived for the moments of Zi(t) and it is shown that for some constants Ni, γ, a, Zi (t)/(Nitγ-1eαt) converges in mean square to a proper random variable.

Random set theory is closely connected with integral geometry. After a general description, based upon the Choquet theorem, the semi-Markovian property is defined and characterized in terms of integral geometry. Applications are made to Poisson polytopes characterized by conditional invariance properties.