We consider a population of reproducing individuals who inherit, earn, consume, and bequeath wealth. A model is constructed to describe the wealth of an individual selected from the nth generation by following a random line of descent from the initial individual. It is shown that bequests are commonly a convex function of wealth. Considering a linear approximation to the bequest function enables us to obtain estimates of the limiting distribution of wealth as the number of generations increases, when earnings of parent and offspring are independent. More generally when earnings of parent and offspring are not independent we obtain upper bounds for the tail of the wealth distribution using a martingale inequality.

Distributional limit theorems, together with rates of convergence, are obtained for the equilibrium distributions of a wide variety of one-dimensional Markov population processes. Three separate cases are considered. First, in the standard setting, the convergence as N→∞ of √N(xN-c) to a normal distribution is established, together with a rate of convergence of O(N−1/2), under weaker conditions than those previously imposed: here, c represents the unique equilibrium of the deterministic equations ẋ = F(x), and xN denotes the population process under its equilibrium distribution. This convergence holds if F′(c)<0: the next section shows that, if F′(c) = 0, both the normalization and the limit distribution are different. Finally, sequences of processes xN suitable for approximating genetical models are considered. In these circumstances, xN itself converges in distribution as N→∞, and the convergence rate is essentially O(N-1), though modification is sometimes needed near natural boundaries.

A two-dimensional linear birth and death process is a continuous-time Markov chain Y(·) with state space (Z+)2 which can jump from the point (n, m) to one of its four neighbors, with rates that are linear functions of n and m. Criteria are extended for determining whether such a process has a positive probability or zero probability of escaping to infinity. In the transient case considered, the projections of the imbedded Markov chain {Xn} of the successive states visited by Y(·) on a suitable pair of orthonormal vectors v and w are shown to be regularly varying sequences with index 1. Specifically, (Xn, v)∽δn and (Xn, w)∽ kn/log n for positive constants δ and k.

If n solid spheres Kn of some volume V(Kn) are scattered randomly in the unit cube of euclidean d-space, some of them will overlap to form Ln(s) molecules with exactly s atoms Kn. The random variable Ln(s) has a limit distribution if V(Kn) tends to zero but nV(Kn) tends to infinity at a certain rate: it is shown that for

Ln(s) is asymptotically Poisson.

Recently, there has been considerable interest in some specialized binary lattice processes. However, this exciting work has been rather fragmentary and heuristic. Rigorous proofs are provided for the existence of some classes of stationary unilateral Markov fields on infinite square lattices. (Neighbours are sites which are horizontally, vertically, or diagonally adjacent.) Many of the difficulties are avoided by characterizing stationary unilateral Markov fields on finite lattices first. Detailed analyses are given for both binary and Gaussian variables.

Let N be a stationary renewal process with a probability density function f(t). Suppose that N can be expressed as the superposition of a finite number of i.i.d. stationary components N(1), …, N(p) (p≧2). Then, under a supplementary condition on f(t), N and N(1), …, N(p) are all Poisson. This is proved by using recurrence relations given in Ito (1978) for the probability distribution of i.i.d. components of a superposition process.

Let X(t) be a homogeneous and continuous stochastic process with independent increments. The subject of this paper is to characterize the stable process by two identically distributed stochastic integrals formed by means of X(t) (in the sense of convergence in probability). The proof of the main results is based on a modern extension of the Phragmén-Lindelöf theory.