We consider d-dimensional stochastic processes which take values in (R+)d These processes generalize Galton-Watson branching processes, but the main assumption of branching processes, independence between particles, is dropped. Instead, we assume for some

Here τ:(R+)d→R +, |x| = σ1d |x(i)|, A {x ∈(R+)d: |x| 1} and T: A→A. Under various assumptions on the maps τ and T it is shown that with probability one there exists a ρ > 1, a fixed point p ∈ A of T and a random variable w such that limn→∞Zn|ρnwp. This result is a generalization of the main limit theorem for supercritical branching processes; note, however, that in the present situation both ρ and ρ are random as well. The results are applied to a population genetical model for zygotic selection without mutation at one locus.

The asymptotic behaviour of those k-type (k ≥ 1) Galton-Watson processes (both with and without immigration) with ρ close to unity is considered. The first principal result (Theorem 3) relates to the vector denoting the numbers of particles of each type at the nth generation in processes without immigration. It is shown that, when normed in a certain way and conditioned on non-extinction, this vector has approximately, for large n, a negative exponential distribution which is degenerate on a line. Theorem 4 is an analogous result for processes with immigration. In this case, no conditioning is required, and the limiting distribution is again degenerate on a line, although now it relates to the gamma rather than the negative exponential.

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.

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.

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.

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.