# Preface

This book provides an extensive introduction to probability and random processes. It is intended for those working in the many and varied applications of the subject as well as for those studying more theoretical aspects. We hope it will be found suitable for mathematics undergraduates at all levels, as well as for graduate students and others with interests in these fields.

In particular, we aim:

• to give a rigorous introduction to probability theory while limiting the amount of measure theory in the early chapters;
• to discuss the most important random processes in some depth, with many examples;
• to include various topics which are suitable for undergraduate courses, but are not routinely taught;
• to impart to the beginner the flavour of more advanced work, thereby whetting the appetite for more.

The ordering and numbering of material in this third edition has for the most part been preserved from the second. However, a good many minor alterations and additions have been made in the pursuit of clearer exposition. Furthermore, we have included new sections on sampling and Markov chain Monte Carlo, coupling and its applications, geometrical probability, spatial Poisson processes, stochastic calculus and the It\^o integral, It\^o's formula and applications, including the Black--Scholes formula, networks of queues, and renewal--reward theorems and applications. In a mild manifestation of millennial mania, the number of exercises and problems has been increased to exceed 1000. These are not merely drill exercises, but complement and illustrate the text, or are entertaining, or (usually, we hope) both. In a companion volume One Thousand Exercises in Probability, (Oxford University Press, 2001), we give worked solutions to almost all exercises and problems.

The basic layout of the book remains unchanged. Chapters 1--5 begin with the foundations of probability theory, move through the elementary properties of random variables, and finish with the weak law of large numbers and the central limit theorem; on route, the reader meets random walks, branching processes, and characteristic functions. This material is suitable for about two lecture courses at a moderately elementary level. The rest of the book is largely concerned with random processes. Chapter 6 deals with Markov chains, treating discrete-time chains in some detail (and including an easy proof of the ergodic theorem for chains with countably infinite state spaces) and treating continuous-time chains largely by example. Chapter 7 contains a general discussion of convergence, together with simple but rigorous accounts of the strong law of large numbers, and martingale convergence. Each of these two chapters could be used as a basis for a lecture course. Chapters 8--13 are more fragmented and provide suitable material for about five shorter lecture courses on: stationary processes and ergodic theory; renewal processes; queues; martingales; diffusions and stochastic integration with applications to finance.

We thank those who have read and commented upon sections of this and earlier editions, and we make special mention of Dominic Welsh, Brian Davies, Tim Brown, Sean Collins, Stephen Suen, Geoff Eagleson, Harry Reuter, David Green, and Bernard Silverman for their contributions to the first edition.

Of great value in the preparation of the second and third editions were the detailed criticisms of Michel Dekking, Frank den Hollander, Torgny Lindvall, and the suggestions of Alan Bain, Erwin Bolthausen, Peter Clifford, Frank Kelly, Doug Kennedy, Colin McDiarmid, and Volker Priebe. Richard Buxton has helped us with classical matters, and Andy Burbanks with the design of the front cover, which depicts a favourite confluence of the authors.

This edition having been reset in its entirety, we would welcome help in thinning the errors should any remain after the excellent \TeX-ing of Sarah Shea-Simonds and Julia Blackwell.

January 2001