ABSTRACT

This paper first presents a review of the connections between various rate of convergence results for Markov chains (including normal Harris ergodicity, geometric ergodicity and sub-geometric rates for a variety of rate functions ψ), and finiteness of appropriate moments of hitting times on small sets. We then present a series of criteria, analogous to Foster's criterion for ergodicity, which imply the finiteness of these moments and hence the rate of convergence of the Markov chain.

These results are applied to random walk on [0, ∞), and we deduce for example that this chain converges at rate ψ(n) = nα (log n)β if the increment variable has finite moment of order nα+1 (log n)β. Similar results hold for more general storage models.

Introduction

It is not always possible to pinpoint the beginnings of a major research area, and yet there can be little doubt that for the bulk of the vast quantity of mathematics known as queueing theory one can trace both the types of problems and the methods of solution to the single fundamental paper of David Kendall [8]. In that paper he introduced the idea of analysing queueing systems using embedded Markov chains, and the interplay between Markov chain theory and queueing theory has since been of vital importance in the development of both areas.

One of the most immediate, simplest and most elegant products of this interplay was the discovery by F.G. Foster (then a student of Kendall) of a criterion for positive recurrence of Markov chains, ensuring that these chains would have stationary distributions.

Introduction

We study in this paper the asymptotics as k → ∞ of the motion of a system of k particles located at sites labelled by the integers. This section gives an informal description of the particle system and our results, and the original motivation for the study.

The particles will be referred to as balls, and the sites as boxes. The motion may be described as follows. Initially the k balls are distributed amongst boxes in such a way that the set of occupied boxes is connected. (A box may contain many balls, but there is no empty box between two occupied boxes.) At each move, a ball is taken from the left-most occupied box and placed one box to the right of a ball chosen uniformly at random from among the k balls, the successive choices being mutually independent. It is clear that the set of occupied boxes remains connected, and that the collection of balls drifts off to infinity. It is easy to see that for each k the k-ball motion drifts off to infinity at an almost certain average speed sk, defined formally by (2.3) below. Our main result is that sk ~ e/k as k → ∞. To be more precise:

THEOREM 1.1 As k increases to infinity, kskincreases to e.

This result was conjectured by Tovey (private communication), and informal arguments supporting the conjecture have been given by Keller (1980) and Weiner (1980). Our method of proof (Sections 2–4) is to use coupling to compare the k-ball process with a certain, more easily analysed, pure growth process (defined at (3.3)).

ABSTRACT

In a discussion of double stochastic population processes in continuous time, attention is concentrated on transition matrices, or equivalent operators, which are linear in the variable parameters. Difficulties with the extreme case of ‘white noise’ variability for the parameters are recalled by reference to ‘stochasticized’ deterministic models, but discussed here also in relation to a general ‘switching’ model.

The use of the ‘backward’ equations for determining extinction probabilities is illustrated by deriving various formulae for the (infinite) birth-and-death process with ‘white-noise’ variability for the birth-and-death coefficients.

INTRODUCTION

It seems very apt to discuss doubly stochastic (d.s.) population processes in this volume, as it recalls the mutual interest of David Kendall and myself in population processes many years ago (e.g. Bartlett, 1947, 1949; Kendall, 1948, 1949; Bartlett and Kendall, 1951). In more recent years d.s. population processes have been considered in discrete time in connection with extinction probabilities for random environments (see, for example, references in Bartlett, 1978, 2.31) and with genetic problems (e.g. Gillespie, 1974); but the concept of d.s processes in continuous time has also become of obvious interest to mathematicians as well as to biologists (e.g. Kaplan, 1973; Keiding, 1975). I might note that my own interest in d.s. processes in continuous time first arose during a visit to Australia in early 1980, when I began to notice in the literature the use of such processes as approximations to genetic or other biological problems for discrete generations.

In my early days as a research student I was both impressed and influenced by David Kendall's enthusiasm for trying out ideas using computer-generated simulations. (His particular project at that time culminated in Kendall, 1974.) This can provide a bridge between what John Tukey calls “exploratory data analysis” and the more classical “confirmatory” aspects of statistics, model estimation and testing. Comparison of data with simulations is definitely “confirmatory” in that models are involved, yet it has much of the spirit of the “exploratory” phase, with human judgement replacing formal significance tests. Later I formalized this comparison to give Monte Carlo tests, discovered earlier by Barnard (1963) but apparently popularized by Ripley (1977).

Simulation has enabled progress to be made in the study of spatial patterns and processes which had previously seemed intractable. The starting point for all known simulation algorithms for spatial point patterns and random sets, such as those in Ripley (1981), is an algorithm to give independent uniformly distributed points in a unit square or cube. The purpose of this paper is to discuss the properties of these basic algorithms. Much of the material can be found scattered in the literature, but no one source gives a sufficiently complete picture.

CONGRUENTIAL RANDOM NUMBER GENERATORS

The usual way to produce approximately uniformly distributed random variables on (0, 1) in a computer is to sample integers xi uniformly from {0,1,…,M–1} or {1,2,…,M–1} and set Ui = xi/M. Here M is a large integer, usually of the form 2β. The approximation made is comparable with that made to represent real numbers by a finite set.

One of the most tattered volumes on my bookshelf is the Symposium on Stochastic Processes, reprinted from the 1949 volume of the Journal of the Royal Statistical Society, Series B. This consists of the three substantial and highly personal survey articles contributed to the symposium by Joseph Moyal, Maurice Bartlett and David Kendall. Together they provided magnificent reading and reference at a time when the stochastic process literature was a meagre scattered one, and continued to do so for many readers over many years. In doing so, they confirmed, if they did not indeed found, the recognisable British tradition in stochastic processes.

David Kendall's article in this volume, “Stochastic processes and population growth”, is only one of many he has written in the course of a distinguished career, and may well not be the one he prizes highest. I single it out, however, both for the seminal effect it had upon me personally, and because it gives an early example of his fascination with the capturing of an applied problem, having both intrinsic interest and an aspect of intangibility, in the meshes of a theory which is light but strong. David's trophies in this benign hunt are legion and living.

Introduction

There have been many models of competition between species, deterministic and stochastic. The most satisfactory of these are those in which the mechanism of competition is made explicit, as competition for limited resources. Deterministic versions of such models lead to the conclusion which O.M. Phillips (1973, 1974) terms the principle of competitive exclusion.

Question for a demy

I hope this salute to my old friend David Kendall will remind him of his Oxford days, for it commemorates an excellent scholarship question that he once set in the 1950s when he was the mathematics tutor at Magdalen. The gist of the question ran as follows. A spherical ball of unit radius rests on an infinite horizontal table. You may imagine that it is a globe with a map of the world painted on its surface to distinguish its spatial orientations. The state of the ball is specified by specifying both its spatial orientation and its position on the table. You have to transfer the ball from a given initial state to an arbitrary final state via a sequence of moves. Each move consists of rolling the ball along some straight line on the table: the length and direction of any move are at your disposal, but the rolling must be pure in the sense that the axis of rotation must be horizontal and there must be no slipping between ball and table. How many moves, N, will be necessary and sufficient to reach any final state?

The original version of the question, set for 18–year–old schoolboys, invited candidates to investigate how two moves, each of length π, would change the ball's orientation; and to deduce in the first place that N ≤ 11, and in the second place that N ≤ 7. Candidates scored bonus marks for any improvement on 7 moves.