After some introductory general remarks on recent investigations involving population models, two broad classes of stochastic model are discussed further, viz. spatial nearest-neighbour lattice models and doubly stochastic models.

In Section 1 of the paper, the first type of model is considered primarily for its relevance to recent work by the author and others affecting the practical design and analysis of replicated field experiments.

In Section 2, doubly stochastic processes are discussed more theoretically, particularly models investigated recently by the author involving infinitesimal transition operators in continuous time linear in the (variable) parameters.

Some new numerical results on extinction and other ‘absorption' probabilities are presented; these are intended to throw further light on the extent to which the assumption of ‘white-noise' variability of the parameters can be a useful approximation to more realistic models.

For the haploid genetic model of Moran, the joint distribution of the numbers of distinct ancestors of a collection of nested subsamples is derived. These results are shown to apply to the diffusion approximations of a wide variety of other genetic models, including the Wright–Fisher process. The results allow us to relate the ancestries of populations sampled at different times. Analogous results for a line-of-descent process that incorporates the effect of mutation are given. Some results about the ages of alleles in an infinite-alleles model are described.

This paper is in three parts. The first deals with the algebraic and topological structure of spaces of rational transfer function linear systems—ARMAX systems, as they have been called. This structure theory is dominated by the concept of a space of systems of order, or McMillan degree, n, because of the fact that this space, M(n), can be realised as a kind of high-dimensional algebraic surface of dimension n(2s + m) where s and m are the numbers of outputs and inputs. In principle, therefore, the fitting of a rational transfer model to data can be considered as the problem of determining n and then the appropriate element of M(n). However, the fact that M(n) appears to need a large number of coordinate neighbourhoods to cover it complicates the task. The problems associated with this program, as well as theory necessary for the analysis of algorithms to carry out aspects of the program, are also discussed in this first part of the paper, Sections 1 and 2.

The second part, Sections 3 and 4, deals with algorithms to carry out the fitting of a model and exhibits these algorithms through simulations and the analysis of real data.

The third part of the paper discusses the asymptotic properties of the algorithm. These properties depend on uniform rates of convergence being established for covariances up to some lag increasing indefinitely with the length of record, T. The necessary limit theorems and the analysis of the algorithms are given in Section 5. Many of these results are of interest independent of the algorithms being studied.

Let v(x, t) denote the displacement of an infinitely long, idealized string performing damped vibrations caused by white noise.

Upper and lower bounds for the distribution of maxsv(x, s) and maxxv(x, t) are presented. The results are obtained by adapting Lévy-type inequalities and exploiting a connection of v(x, t) with the Ornstein-Uhlenbeck process through Slepian's theorem.

The case of forced-damped vibrations is also analysed. Finally, a section is devoted to the case of a semi-infinite string performing damped vibrations.

We consider the statistical estimation problem of potential functions of Gibbs models on the plane lattice. We assume that the area on which a random point pattern is observed is sufficiently large and take an asymptotic point of view. The main result is to show the locally asymptotic normality of the Gibbs model under certain conditions. From this result we can show the optimality of the maximum likelihood estimator employing known results about locally asymptotic normal families, though a practical computation of the maximum likelihood estimator presents difficulties. An estimation procedure based on the method of moments is also proposed.

The asymptotic moments of the concentration of a pollutant subject to rainout and dry removal are given. Both the lengths of the wet and dry periods and the wet and dry removal rates are allowed to be random, although the removal rate is assumed to be fixed within any one wet or dry period. The results hold in a more general setting than removal processes, so its is hoped that they will be applicable to other problems.

Let K be a finite graph with vertex set V = {x0, x1, …, xσ–1} and automorphism group G. It is assumed that G acts transitively on V. We can imagine that the vertices of K represent σ cities and a traveler visits the cities in a series of random flights. The traveler starts at a given city and in each flight, independently of the past journey, chooses a city at random as the destination. Denote by vn (n = 1, 2, …) the location of the traveler at the end of the nth flight, and by v0 the initial location. It is assumed that the transition probabilities P{vn = xj | vn–1 = xi}, xi ϵ V, xj ϵ V, do not depend on n and are invariant under the action of G on V. The main result of this paper consists in determining p(n), the probability that the traveler returns to the initial position at the end of the nth flight.

A body receives a sequence of rotations through a fixed angle about an axis whose direction is arbitrary. The probability distribution governing the resulting orientation of the body is determined. The problem is generalized to the case where the axis of each individual rotation makes the same angle, Θ, with an axis fixed in the body but is otherwise random. The resulting distribution is shown, in the case , to reduce to the Roberts-Ursell distribution for random walk on a sphere. Some diffusion limits are examined.

We give here concrete formulas relating the transition generatrix functions of any random walk on a finite group to the irreducible representations of this group. Some examples of such explicit calculations for the permutation groups A4, S4, and A5 are included.

Let with Z(0) = 0 be a random process under investigation and N be a point process associated with Z. Both Z and N are defined on the same probability space. Let with R0 = 0 denote the consecutive positions of points of N on the half-line . In this paper we present sufficient conditions under which (Z, R) is a new better than used (NBU) process and give several examples of NBU processes satisfying these conditions. In particular we consider the processes in which N is a renewal and a general point process. The NBU property of some semi-Markov processes is also presented.

Known results for the two-sex birth-death population process are extended to the case where immigration of females is allowed. It is shown that the methods used follow those for the standard birth, death and immigration process.

Consider bond-percolation on a graph G with sites S(G). We disprove the conjecture of Hammersley (1957) that the function n → sups ϵ S(G)E [the number of sites s′ at distance n from s which can be reached from s by an open path which, except for s′, only passes through sites at distance smaller than n from s] is always subexponential.

In a recent letter, Deshpande and Kochar (1983) gave the following theorem.

Shaked (1982) gives a characterisation of dispersive ordering for distributions which are absolutely continuous with interval supports. However there is a technical error in his proof of the second part of Theorem 2.1, although the result is still true. The difficulty arises because zero terms are discarded in the definition of the sign change operator S. Therefore the equality of and x′ does not imply that

We consider the flow of service starting times in the M/G/1/∞ queue and study some of its equilibrium properties.