This article attempts to provide a historical perspective of the author's involvement in applied probability. Several research problems in this area are described, and three organizational experiences of importance to the author outlined. After a brief account of the development of the Applied Probability journals, the article concludes with some philosophical remarks.

After some reminiscences about the founding of the Applied Probability journals the author discusses three open problems concerned respectively with:

(i) the prehistory of applied probability

(ii) Poisson–Delauney Simplexes

(iii) voids and pancakes in the universe.

This is a discussion of the place of empirical methods in the study of probability models. It is argued that, for models of great complexity, careful experimentation may be a legitimate source of factual information on the behavior of such models. The need for rigorous standards of empirical methodology and of appropriate reporting is stressed. A simple, but interesting question on the distribution of the maximal eigenvalue of positive matrices with random elements is used as an example of an experimental study. In the course of the discussion of that example, several operational rules of sound computer experimentation are stated.

An account is given of the foundation of the Applied Probability Trust and the development of its office at the University of Sheffield. The publishing achievements of its first quarter-century are reviewed.

The effects of computer technology on various aspects of mathematical printing are discussed. A brief account is given of the use of a microcomputer and commercial software in a publishing office environment.

A survey of the development of statistical techniques of field experimentation refers to the unified approach of R. A. Fisher, in which analysis and design were closely linked. Randomization of treatment positions bypassed the need for a precise stochastic model, but sometimes at the cost of diminished accuracy.

Alternative covariance methods associated with the name of J. S. Papadakis, taking note of the yields of neighbouring plots, have stimulated much research in recent years. The results to date suggest that, while improved accuracy can often be achieved, the assessment of accuracy is more dependent on the stochastic model. Robust methods for such assessment still need to be finalized; with these alternative methods a closer link between analysis and design is also advocated.

We discuss the role of stochastic processes in modelling the life-cycle of a biological cell and the growth of cell populations. Results for multiphase age-dependent branching processes have proved invaluable for the interpretation of many of the basic experimental studies of the life-cycle. Moreover problems from cell kinetics, in particular those related to diurnal rhythm in cell-growth, have stimulated research into ‘periodic' renewal theory, and the asymptotic behaviour of populations of cells with periodic death rate.

It is known that several different alleles can be maintained at a locus by selection, but only when the various genotypic fitnesses satisfy very special conditions. It is shown in this paper that a population with many possible mutations will evolve in such a way that these conditions arise naturally for most fitness regimes. The first steps are taken towards the assessment of the likely size and shape of the resulting stable polymorphisms.

Some simple methods of estimating inbreeding from tests on a random sample of individuals are discussed. These are mostly iterative procedures, converging to the maximum likelihood estimate, with algorithms for finding standard errors. But there are problems of interpretation, both from the statistical and biological points of view.

Problems associated with setting up a probability model which generates the quantum theoretical probabilities for the two spin 1/2 particle system are examined. Arguments which claim to show that such a model cannot be constructed within classical probability theory under the assumption of local singlet states are also considered. It is shown that the model then in question is not a probability model in the sense that term is used elsewhere in science. An alternative model is proposed and its bearing on the Einstein-Bohr debate is briefly discussed.

The notion of a quantum random walk in discrete time is formulated and the passage to a continuous time diffusion limit is established. The limiting diffusion is described in terms of solutions of certain quantum stochastic differential equations.

Five distributions on the non-negative integers are given which use the orthogonal polynomials of Hermite, Legendre, Laguerre and Chebyshev.

Stein's method of obtaining rates of convergence, well known in normal and Poisson approximation, is considered here in the context of approximation by Poisson point processes, rather than their one-dimensional distributions. A general technique is sketched, whereby the basic ingredients necessary for the application of Stein's method may be derived, and this is applied to a simple problem in Poisson point process approximation.

For a strong Markov process on the line with continuous paths the Karlin–McGregor determinant formula of coincidence probabilities for multiple particle systems is extended to allow the individual component processes to start at variable times and run for variable durations. The extended formula is applied to a variety of combinatorial problems including counts of non-crossing paths in the plane with variable start and end points, dominance orderings, numbers of dominated majorization orderings, and time-inhomogeneous random walks.

Results showing the weak convergence of certain stochastic processes are used to derive both known and new (asymptotic) properties of signs of residuals from regression; other weak convergence results are derived, and used to determine the behaviour of runs of residuals.

Let W be the usual almost-sure limit random variable in a supercritical simple branching process; we study its tail behaviour. For the left tail, we distinguish two cases, the ‘Schröder' and ‘Böttcher' cases; both appear in work of Harris and Dubuc. The Schröder case is related to work of Karlin and McGregor on embeddability in continuous-time (Markov) branching processes. New results are obtained for the Böttcher case; there are links with recent work of Barlow and Perkins on Brownian motion on a fractal. The right tail is also considered. Use is made of recent progress in Tauberian theory.

By Ignatov's theorem the sets of values in an i.i.d. sequence that are the kth largest at their appearance (k = 1, 2, ···) are supports of i.i.d. Poisson processes. The present paper contains an elementary and short proof for the case where the underlying distribution function F is discrete, and then extends the result to general F.

The matrix-geometric results of M. Neuts are extended to ergodic row-continuous bivariate Markov processes [J(t), N(t)] on state space B = {(j, n)} for which: (a) there is a boundary level N for N(t) associated with finite buffer capacity; (b) transition rates to adjacent rows and columns are independent of row level n in the interior of B. Such processes are of interest in the modelling of queue-length for voice-data transmission in communication systems.

One finds that the ergodic distribution consists of two decaying components of matrix-geometric form, the second induced by the finite buffer capacity. The results are obtained via Green's function methods and compensation. Passage-time distributions for the two boundary problems are also made available algorithmically.

We develop a theory of semiregenerative phenomena. These may be viewed as a family of linked regenerative phenomena, for which Kingman [6], [7] developed a theory within the framework of quasi-Markov chains. We use a different approach and explore the correspondence between semiregenerative sets and the range of a Markov subordinator with a unit drift (or a Markov renewal process in the discrete-time case). We use techniques based on results from Markov renewal theory.

Foster's criterion for positive recurrence of irreducible countable space Markov chains is one of the oldest tools in applied probability theory. In various papers in JAP and AAP it has been shown that, under extensions of irreducibility such as ϕ -irreducibility, analogues of and generalizations of Foster's criterion give conditions for the existence of an invariant measure π for general space chains, and for π to have a finite f-moment ∫π (dy)f(y), where f is a general function. In the case f ≡ 1 these cover the question of finiteness of π itself.

In this paper we show that the same conditions imply the same conclusions without any irreducibility assumptions; Foster's criterion forces sufficient and appropriate regularity on the space automatically. The proofs involve detailed consideration of the structure of the minimal subinvariant measures of the chain.

The results are applied to random coefficient autoregressive processes in order to illustrate the need to remove irreducibility conditions if possible.