An introduction to the theory of point processes

The rest of the book has been revised to take these changes into account, to correcterrors in the first edition, and to bring in a range of new ideas and examples.Even at the time of the

Theory of Point Processes: Volume I: Elementary Theory and Methods, Second Edition

D.J Daley

D Vere-Jones

Springer

A Series of the Applied Probability Trust

Editors: J Gani, C.C Heyde, T.G Kurtz

An Introduction to the

Theory of Point Processes

Volume I: Elementary Theory and Methods

Second Edition

Applications Computing Sciences

Mathematical Sciences Institute Victoria University of WellingtonAustralian National University Wellington, New Zealand

Canberra, ACT 0200, Australia David.Vere-Jones@mcs.vuw.ac.nzdaryl@maths.anu.edu.au

Series Editors:

Australian National Australian National University of Wisconsin

Library of Congress Cataloging-in-Publication Data

Daley, Daryl J.

An introduction to the theory of point processes / D.J Daley, D Vere-Jones.

p cm.

Includes bibliographical references and index.

Contents: v 1 Elementary theory and methods

ISBN 0-387-95541-0 (alk paper)

1 Point processes I Vere-Jones, D (David) II Title

QA274.42.D35 2002

ISBN 0-387-95541-0 Printed on acid-free paper.

© 2003, 1988 by the Applied Probability Trust.

All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York,

NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer soft- ware, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed in the United States of America.

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Typesetting: Photocomposed pages prepared by the authors using plain TeX files.

www.springer-ny.com

Springer-Verlag New York Berlin Heidelberg

A member of BertelsmannSpringer Science+Business Media GmbH

To Nola,

and in memory of Mary

◆

In preparing this second edition, we have taken the opportunity to reshapethe book, partly in response to the further explosion of material on pointprocesses that has occurred in the last decade but partly also in the hope

of making some of the material in later chapters of the first edition moreaccessible to readers primarily interested in models and applications Topicssuch as conditional intensities and spatial processes, which appeared relativelyadvanced and technically difficult at the time of the first edition, have nowbeen so extensively used and developed that they warrant inclusion in theearlier introductory part of the text Although the original aim of the book—

to present an introduction to the theory in as broad a manner as we areable—has remained unchanged, it now seems to us best accomplished in twovolumes, the first concentrating on introductory material and models and thesecond on structure and general theory The major revisions in this volume,

as well as the main new material, are to be found in Chapters 6–8 The rest

of the book has been revised to take these changes into account, to correcterrors in the first edition, and to bring in a range of new ideas and examples.Even at the time of the first edition, we were struggling to do justice tothe variety of directions, applications and links with other material that thetheory of point processes had acquired The situation now is a great dealmore daunting The mathematical ideas, particularly the links to statisticalmechanics and with regard to inference for point processes, have extendedconsiderably Simulation and related computational methods have developedeven more rapidly, transforming the range and nature of the problems underactive investigation and development Applications to spatial point patterns,especially in connection with image analysis but also in many other scien-tific disciplines, have also exploded, frequently acquiring special language andtechniques in the different fields of application Marked point processes, whichwere clamouring for greater attention even at the time of the first edition, haveacquired a central position in many of these new applications, influencing boththe direction of growth and the centre of gravity of the theory

vii

We are sadly conscious of our inability to do justice to this wealth of newmaterial Even less than at the time of the first edition can the book claim toprovide a comprehensive, up-to-the-minute treatment of the subject Nor are

we able to provide more than a sketch of how the ideas of the subject haveevolved Nevertheless, we hope that the attempt to provide an introduction

to the main lines of development, backed by a succinct yet rigorous treatment

of the theory, will prove of value to readers in both theoretical and appliedfields and a possible starting point for the development of lecture courses ondifferent facets of the subject As with the first edition, we have endeavoured

to make the material as self-contained as possible, with references to ground mathematical concepts summarized in the appendices, which appear

back-in this edition at the end of Volume I

We would like to express our gratitude to the readers who drew our tention to some of the major errors and omissions of the first edition andwill be glad to receive similar notice of those that remain or have been newlyintroduced Space precludes our listing these many helpers, but we would like

at-to acknowledge our indebtedness at-to Rick Schoenberg, Robin Milne, VolkerSchmidt, G¨unter Last, Peter Glynn, Olav Kallenberg, Martin Kalinke, JimPitman, Tim Brown and Steve Evans for particular comments and carefulreading of the original or revised texts (or both) Finally, it is a pleasure tothank John Kimmel of Springer-Verlag for his patience and encouragement,and especially Eileen Dallwitz for undertaking the painful task of rekeying thetext of the first edition

The support of our two universities has been as unflagging for this our as for the first edition; we would add thanks to host institutions of visits

endeav-to the Technical University of Munich (supported by a Humboldt FoundationAward), University College London (supported by a grant from the Engineer-ing and Physical Sciences Research Council) and the Institute of Mathematicsand its Applications at the University of Minnesota

This book has developed over many years—too many, as our colleagues andfamilies would doubtless aver It was conceived as a sequel to the review paperthat we wrote for the Point Process Conference organized by Peter Lewis in

1971 Since that time the subject has kept running away from us faster than

we could organize our attempts to set it down on paper The last two decadeshave seen the rise and rapid development of martingale methods, the surge

of interest in stochastic geometry following Rollo Davidson’s work, and theforging of close links between point processes and equilibrium problems instatistical mechanics

Our intention at the beginning was to write a text that would provide a

survey of point process theory accessible to beginning graduate students and

workers in applied fields With this in mind we adopted a partly historicalapproach, starting with an informal introduction followed by a more detaileddiscussion of the most familiar and important examples, and then movinggradually into topics of increased abstraction and generality This is still thebasic pattern of the book Chapters 1–4 provide historical background andtreat fundamental special cases (Poisson processes, stationary processes onthe line, and renewal processes) Chapter 5, on finite point processes, has abridging character, while Chapters 6–14 develop aspects of the general theory.The main difficulty we had with this approach was to decide when andhow far to introduce the abstract concepts of functional analysis With someregret, we finally decided that it was idle to pretend that a general treatment ofpoint processes could be developed without this background, mainly becausethe problems of existence and convergence lead inexorably to the theory ofmeasures on metric spaces This being so, one might as well take advantage

of the metric space framework from the outset and let the point process itself

be defined on a space of this character: at least this obviates the tedium ofhaving continually to specify the dimensions of the Euclidean space, while inthe context of completely separable metric spaces—and this is the greatest

ix

generality we contemplate—intuitive spatial notions still provide a reasonableguide to basic properties For these reasons the general results from Chapter

6 onward are couched in the language of this setting, although the examplescontinue to be drawn mainly from the one- or two-dimensional EuclideanspacesR1 andR2 Two appendices collect together the main results we needfrom measure theory and the theory of measures on metric spaces We hopethat their inclusion will help to make the book more readily usable by appliedworkers who wish to understand the main ideas of the general theory withoutthemselves becoming experts in these fields Chapter 13, on the martingaleapproach, is a special case Here the context is again the real line, but weadded a third appendix that attempts to summarize the main ideas neededfrom martingale theory and the general theory of processes Such specialtreatment seems to us warranted by the exceptional importance of these ideas

in handling the problems of inference for point processes

In style, our guiding star has been the texts of Feller, however many years we may be from achieving that goal In particular, we have tried tofollow his format of motivating and illustrating the general theory with arange of examples, sometimes didactical in character, but more often takenfrom real applications of importance In this sense we have tried to strike

light-a melight-an between the rigorous, light-abstrlight-act trelight-atments of texts such light-as those byMatthes, Kerstan and Mecke (1974/1978/1982) and Kallenberg (1975, 1983),and practically motivated but informal treatments such as Cox and Lewis(1966) and Cox and Isham (1980)

Numbering Conventions Each chapter is divided into sections, with

con-secutive labelling within each of equations, statements (encompassing tions, Conditions, Lemmas, Propositions, Theorems), examples, and the ex-ercises collected at the end of each section Thus, in Section 1.2, (1.2.3) is the

Defini-third equation, Statement 1.2.III is the Defini-third statement,Example 1.2(c)

is the third example, and Exercise 1.2.3 is the third exercise The exercisesare varied in both content and intention and form a significant part of thetext Usually, they indicate extensions or applications (or both) of the theoryand examples developed in the main text, elaborated by hints or referencesintended to help the reader seeking to make use of them The symbol de-notes the end of a proof Instead of a name index, the listed references carrypage number(s) where they are cited A general outline of the notation usedhas been included before the main text

It remains to acknowledge our indebtedness to many persons and tions Any reader familiar with the development of point process theory overthe last two decades will have no difficulty in appreciating our dependence onthe fundamental monographs already noted by Matthes, Kerstan and Mecke

institu-in its three editions (our use of the abbreviation MKM for the 1978 Englishedition is as much a mark of respect as convenience) and Kallenberg in itstwo editions We have been very conscious of their generous interest in ourefforts from the outset and are grateful to Olav Kallenberg in particular forsaving us from some major blunders A number of other colleagues, notably

David Brillinger, David Cox, Klaus Krickeberg, Robin Milne, Dietrich Stoyan,Mark Westcott, and Deng Yonglu, have also provided valuable comments andadvice for which we are very grateful Our two universities have respondedgenerously with seemingly unending streams of requests to visit one another

at various stages during more intensive periods of writing the manuscript Wealso note visits to the University of California at Berkeley, to the Center forStochastic Processes at the University of North Carolina at Chapel Hill, and

to Zhongshan University at Guangzhou For secretarial assistance we wish

to thank particularly Beryl Cranston, Sue Watson, June Wilson, Ann gan, and Shelley Carlyle for their excellent and painstaking typing of difficultmanuscript

Milli-Finally, we must acknowledge the long-enduring support of our families,and especially our wives, throughout: they are not alone in welcoming thespeed and efficiency of Springer-Verlag in completing this project

Preface to the Second Edition vii

2 Basic Properties of the Poisson Process 19

2.2 Characterizations of the Stationary Poisson Process:

2.3 Characterizations of the Stationary Poisson Process:

3 Simple Results for Stationary Point Processes on the Line 41

3.3 Mean Density, Intensity, and Batch-Size Distribution 46

3.5 Ergodicity and an Elementary Renewal Theorem Analogue 60

xiii

4 Renewal Processes 66

4.5 Neighbours of the Renewal Process: Wold Processes 92

4.6 Stieltjes-Integral Calculus and Hazard Measures 106

5 Finite Point Processes 111

5.1 An Elementary Example: Independently and Identically

5.2 Factorial Moments, Cumulants, and Generating Function

5.3 The General Finite Point Process: Definitions and Distributions 123

6 Models Constructed via Conditioning:

Cox, Cluster, and Marked Point Processes 157

7 Conditional Intensities and Likelihoods 211

7.2 Conditional Intensities, Likelihoods, and Compensators 229

7.3 Conditional Intensities for Marked Point Processes 246

8 Second-Order Properties of Stationary Point Processes 288

A1 A Review of Some Basic Concepts of

Topology and Measure Theory 368

A2 Measures on Metric Spaces 384

A2.6 Boundedly Finite Measures and the Space M#

A3 Conditional Expectations, Stopping Times,

Chapter Titles for Volume II

9 General Theory of Point Processes and Random Measures

10 Special Classes of Processes

11 Convergence Concepts and Limit Theorems

12 Ergodic Theory and Stationary Processes

13 Palm Theory

14 Evolutionary Processes and Predictability

15 Spatial Point Processes

Very little of the general notation used in Appendices 1–3 is given below Also,notation that is largely confined to one or two sections of the same chapter

is mostly excluded, so that neither all the symbols used nor all the uses ofthe symbols shown are given The repeated use of some symbols occurs as aresult of point process theory embracing a variety of topics from the theory ofstochastic processes Where they are given, page numbers indicate the first

or significant use of the notation Generally, the particular interpretation ofsymbols with more than one use is clear from the context

Throughout the lists below, N denotes a point process and ξ denotes a

X state space of N or ξ; often X = R d; alwaysX is

c.s.m.s (complete separable metric space)

Ω space of probability elements ω

∅, ∅(·) null set, null measure

E measurable sets in probability space

(Ω, E, P) basic probability space on which N and ξ are defined 158

xvii

B(X ) Borel σ-field generated by open spheres of

B (n)

X =B(X (n)) product σ-field on product space X (n) 129

BM+(X ) measurable nonnegative functions of bounded

M X (N X) totally finite (counting) measures on c.s.m.s.X 158, 398

M#

N#

U ⊗ V product topology on product space X × Y of

V = V(X ) [0, 1]-valued measurable functions h(x) with

φ = Fourier transform of Lebesgue integrable

# extension of concept from totally finite to

a.e µ, µ-a.e almost everywhere with respect to measure µ 376

A family of sets generatingB; semiring of

bounded Borel sets generatingB X 31, 368

B u (T u) backward (forward) recurrence time at u 58, 76

c k , c [k] kth cumulant, kth factorial cumulant,

c(x) = c(y, y + x)

covariance density of stationary mean square

C [k](·), c [k](·) factorial cumulant measure and density 147

˘

C2(·), ˘c(·) reduced covariance measure of stationary N or ξ 292

˘

c( ·) reduced covariance density of stationary N or ξ 160, 292

F n ∗ n-fold convolution power of measure or d.f F 55

F ( · ; ·) finite-dimensional (fidi) distribution 158–161

G[h] probability generating functional (p.g.fl.) of N , 15, 144

G c[·], G m[· | x] p.g.fl.s of cluster centre and cluster member

I A (x) = δ x (A) indicator function of element x in set A

L u = B u + T u current lifetime of point process onR 58, 76

L[f ] (f ∈ BM+(X ))

L2(ξ0), L2(Γ) Hilbert spaces of square integrable r.v.s ξ0, and

of functions square integrable w.r.t measure Γ 332

L A (x1, , x n ), = j N (x1, , x N | A)

likelihood, local Janossy density, N ≡ N(A) 22, 212

m k (m [k]) kth (factorial) moment of distribution {p n } 115

m2, ˘ M2 reduced second-order moment density, measure,

mg mean density of ground process Ng of MPP N 198, 323

N (a, b] number of points in half-open interval (a, b], 19

{(p n , Π n)} elements of probability measure for

P (z) probability generating function (p.g.f.) of

Q( ·), Q(t) hazard measure, integrated hazard function (IHF) 109

renewal function (U (x) = 1 + U0(x)) 67

V (x) = V ((0, x]) variance function for stationary N or ξ onR 80, 301

11.2.I–II 8.2.I–II11.3.I–VIII 8.4.I–VIII

13.1.I–III 7.1.I–III13.1.IV–VI 7.2.I–III

A1.1.I–5.IV A1.1.I–5.IVA2.1.I–III A2.1.I–III

A2.2.I–7.III A2.2.I–7.IIIA3.1.I–4.IX A3.1.I–4.IX

xxi

Early History

The ancient origins of the modern theory of point processes are not easy totrace, nor is it our aim to give here an account with claims to being definitive.But any retrospective survey of a subject must inevitably give some focus onthose past activities that can be seen to embody concepts in common with themodern theory Accordingly, this first chapter is a historical indulgence butwith the added benefit of describing certain fundamental concepts informallyand in a heuristic fashion prior to possibly obscuring them with a plethora ofmathematical jargon and techniques These essentially simple ideas appear

to have emerged from four distinguishable strands of enquiry—although ourdivision of material may sometimes be a little arbitrary These are

(i) life tables and the theory of self-renewing aggregates;

(ii) counting problems;

(iii) particle physics and population processes; and

(iv) communication engineering

The first two of these strands could have been discerned in centuries pastand are discussed in the first two sections The remaining two essentiallybelong to the twentieth century, and our comments are briefer in the remainingsection

Of all the threads that are woven into the modern theory of point processes,the one with the longest history is that associated with intervals betweenevents This includes, in particular, renewal theory, which could be defined

in a narrow sense as the study of the sequence of intervals between successivereplacements of a component that is liable to failure and is replaced by a new

1

component every time a failure occurs As such, it is a subject that oped during the 1930s and reached a definitive stage with the work of Feller,Smith, and others in the period following World War II But its roots extendback much further than this, through the study of ‘self-renewing aggregates’

devel-to problems of statistical demography, insurance, and mortality tables—inshort, to one of the founding impulses of probability theory itself It is noteasy to point with confidence to any intermediate stage in this chronicle thatrecommends itself as the natural starting point either of renewal theory or ofpoint process theory more generally Accordingly, we start from the begin-ning, with a brief discussion of life tables themselves The connection withpoint processes may seem distant at first sight, but in fact the theory of lifetables provides not only the source of much current terminology but also thesetting for a range of problems concerning the evolution of populations intime and space, which, in their full complexity, are only now coming withinthe scope of current mathematical techniques

In its basic form, a life table consists of a list of the number of individuals,usually from an initial group of 1000 individuals so that the numbers areeffectively proportions, who survive to a given age in a given population

The most important parameters are the number x surviving to age x, the number d x dying between the ages x and x + 1 (d x = x − x+1), and the

number q x of those surviving to age x who die before reaching age x + 1 (q x = d x / x) In practice, the tables are given for discrete ages, with theunit of time usually taken as 1 year For our purposes, it is more appropriate

to replace the discrete time parameter by a continuous one and to replace

numbers by probabilities for a single individual Corresponding to xwe havethen the survivor function

S(x) = Pr {lifetime > x}.

To d x corresponds f (x), the density of the lifetime distribution function, where

f (x) dx = Pr {lifetime terminates between x and x + dx},

while to q x corresponds q(x), the hazard function, where

q(x) dx = Pr {lifetime terminates between x and x + dx

| it does not terminate before x.}

Denoting the lifetime distribution function itself by F (x), we have the

follow-ing important relations between the functions above:

The first life table appeared, in a rather crude form, in John Graunt’s (1662)

Observations on the London Bills of Mortality This work is a landmark in the

early history of statistics, much as the famous correspondence between Pascaland Fermat, which took place in 1654 but was not published until 1679, is

a landmark in the early history of formal probability The coincidence indates lends weight to the thesis (see e.g Maistrov, 1967) that mathematicalscholars studied games of chance not only for their own interest but for theopportunity they gave for clarifying the basic notions of chance, frequency, andexpectation, already actively in use in mortality, insurance, and populationmovement contexts

An improved life table was constructed in 1693 by the astronomer Halley,using data from the smaller city of Breslau, which was not subject to thesame problems of disease, immigration, and incomplete records with whichGraunt struggled in the London data Graunt’s table was also discussed byHuyghens (1629–1695), to whom the notion of expected length of life is due

A de Moivre (1667–1754) suggested that for human populations the function

S(x) could be taken to decrease with equal yearly decrements between the ages

22 and 86 This corresponds to a uniform density over this period and a hazard

function that increases to infinity as x approaches 86 The analysis leading

to (1.1.1) and (1.1.2), with further elaborations to take into account differentsources of mortality, would appear to be due to Laplace (1747–1829) It is

interesting that in A Philosophical Essay on Probabilities (1814), where the

classical definition of probability based on equiprobable events is laid down,Laplace gave a discussion of mortality tables in terms of probabilities of atotally different kind Euler (1707–1783) also studied a variety of problems ofstatistical demography

From the mathematical point of view, the paradigm distribution functionfor lifetimes is the exponential function, which has a constant hazard inde-

pendent of age: for x > 0, we have

f (x) = λe −λx , q(x) = λ, S(x) = e −λx , F (x) = 1 − e −λx . (1.1.4)

The usefulness of this distribution, particularly as an approximation for poses of interpolation, was stressed by Gompertz (1779–1865), who also sug-gested, as a closer approximation, the distribution function corresponding to

pur-a power-lpur-aw hpur-azpur-ard of the form

q(x) = Ae αx (A > 0, α > 0, x > 0) (1.1.5) With the addition of a further constant [i.e q(x) = B + Ae αx], this is known

in demography as the Gompertz–Makeham law and is possibly still the most

widely used function for interpolating or graduating a life table

Other forms commonly used for modelling the lifetime distribution in

dif-ferent contexts are the Weibull, gamma, and log normal distributions,

corre-sponding, respectively, to the formulae

q(x) = βλx β −1 with S(x) = exp( −λx β) (λ > 0, β > 0), (1.1.6)

be emphasized that as a general rule the same distribution may arise fromseveral models (see Exercise 1.1.3).

The gamma distribution has a long history and arises in many different

contexts When α = 12k and λ = 12, it is nothing other than the chi-squared

distribution with k degrees of freedom, with well-known applications in ematical statistics When α = 1, it reduces to the exponential distribution, and when α = 32, it reduces to the Maxwell distribution for the distribution

math-of energies math-of molecules in a perfect gas The most important special cases

in the context of life tables arise when α is a positive integer, say α = k.

It then has an interpretation as the sum of k independent random variables,

each having an exponential distribution Although commonly known as theErlang distribution, after the Danish engineer and mathematician who intro-duced it as a model for telephone service and intercall distributions in the1920s, this special form and its derivation were known much earlier One ofthe earliest derivations, if not the first, is due to the English mathematicianR.C Ellis (1817–1859) in a remarkable paper in 1844 that could well be hailed

as one of the early landmarks in stochastic process theory, although in fact

it is rarely quoted In addition to establishing the above-mentioned result

as a special case, Ellis studied a general renewal process and in that contextestablished the asymptotic normality of the sum of a number of independentnonnegative random variables It is particularly remarkable in that he usedFourier methods; in other words, essentially the modern characteristic func-tion proof (with a few lacunae from a modern standpoint) of the central limittheorem

An equally interesting aspect of Ellis’ paper is the problem that inspiredthe study This takes us back a century and a half to an even less familiarstatistician in the guise of Sir Isaac Newton (1642–1728) For much of hislater life, Newton’s spare time was devoted to theological problems, one ofwhich was to reconcile the ancient Greek and Hebrew chronologies In bothchronologies, periods of unknown length are spanned by a list of successiverulers Newton proposed to estimate such periods, and hence to relate thetwo chronologies, by supposing each ruler to reign for a standard period of 22years This figure was obtained by a judicious comparison of averages from amiscellany of historical data for which more or less reliable lengths of reignswere known It is a statistical inference in the same sense as many of Graunt’sinferences from the London Bills of Mortality: a plausible value based on thebest or only evidence available and supported by as many cross-checks ascan be devised How far it was explicitly present in Newton’s mind that hewas dealing with a statistical problem and whether he made any attempts

to assess the likely errors of his results himself are questions we have notbeen able to answer with any certainty In an informal summary of his work,Newton (1728) wrote: “I do not pretend to be exact to a year: there may

be errors of five or ten years, and sometimes twenty, and not much above.”However, it appears unlikely that these figures were obtained by any theory

of compounding of errors It is tempting to conjecture that he may havediscussed the problems with such friends and Fellows of the Royal Society asHalley, whose paper to the Royal Society would have been presented while

Newton was president, and de Moivre, who dedicated the first edition of The Doctrine of Chances to Newton, but if records of such discussions exist, we

have not found them

Up until the middle of the nineteenth century, as will be clear even fromthe brief review presented above, mathematical problems deriving from life ta-bles not only occupied a major place in the subject matter of probability andstatistics but also attracted the attention of many leading mathematicians ofthe time From the middle of the nineteenth century onward, however, actu-arial mathematics (together, it may be added, with many other probabilisticnotions), while important in providing employment for mathematicians, be-came somewhat disreputable mathematically, a situation from which it hasnot fully recovered (How many elementary textbooks in statistics, for ex-ample, even mention life tables, let alone such useful descriptive tools as thehazard function?) The result was that when, as was inevitably the case, newapplications arose that made use of the same basic concepts, the links withearlier work were lost or only partially recognized Moreover, the new de-velopments themselves often took place independently or with only a partialrealization of the extent of common material

In the twentieth century, at least three such areas of application may bedistinguished The first, historically, was queueing theory, more specificallythe theory of telephone trunking problems Erlang’s (1909) first paper onthis subject contains a derivation of the Poisson distribution for the number

of calls in a fixed time interval It is evident from his comments that evenbefore that time the possibility of using probabilistic methods in that contextwas being considered by engineers in several countries The work here appears

to be quite independent of earlier contributions In later work, the analysiswas extended to cover queueing systems with more general input and servicedistributions

Mathematical interest in actuarial problems as such re-emerged in the 1910sand 1920s in connection with the differential and integral equations of popu-lation growth Here at least there is a bridge between the classical theory oflife tables on the one hand and the modern treatments of renewal processes onthe other It is provided by the theory of ‘self-renewing aggregates’ [to borrow

a phrase from the review by Lotka (1939), which provides a useful survey ofearly work in this field], a term that refers to a population (portfolio in theinsurance context) of individuals subject to death but also able to regeneratethemselves so that a stable population can be achieved

As a typical illustration, consider the evolution of a human population for

which it is assumed that each female of age x has a probability φ(x) dt of giving birth to a daughter in a time interval of length dt, independently of the

behaviour of other females in the population and also of any previous children

she may have had Let S(x) denote the survivor function for the (female) life distribution and n(t) the expected female birth rate at time t Then n(t)

satisfies the integral equation

n(t) =

 t

0

n(t − x)S(x)φ(x) dx,

which represents a breakdown of the total female birth rate by age of parent

If the population is started at time zero with an initial age distribution having

density r(x), the equation can be rewritten in the form

is the contribution to the birth rate at time t from the initial population In

this form, the analogy with the integral equation of renewal theory is clear.Indeed, the latter equation corresponds to the special case where at death eachindividual is replaced by another of age zero and no other ‘births’ are possible.The population size then remains constant, and it is enough to consider a

population with just one member In place of n(t), we then have the renewal density m(t), with m(t) dt representing the probability that a replacement will be required in the small time interval (t, t + dt); also, φ(x) becomes the hazard function h(x) for the life distribution, and the combination S(x)h(x) can be replaced by the probability density function f (x) as in (1.1.3) Thus,

we obtain the renewal equation in the form

m(t) = n0(t) +

 t

0

m(t − u)f(u) du.

If, finally, the process is started with a new component in place at time 0,

then n0(t) = f (t) and we have the standard form

of lifetime distributions of systems of elements connected in series (‘weakestlink’ model) or in parallel Weibull’s analysis is an example of the first type ofmodel, which typically leads to an extreme-value distribution with a long righttail An early example of a parallel model is Daniels’ (1945) treatment of thefailure of fibre bundles; the distributions in this case have an asymptoticallynormal character In between and extending these two primary cases lie aninfinite variety of further failure models, in all of which the concepts andterminology invented to cover the life table problem play a central role.

In retrospect, it is easy to see that the three fields referred to are closelyinterconnected Together, they provide one of the main areas of applicationand development of point process theory Of course, they do not represent theonly fields where life table methods have been applied with success An earlypaper by Watanabe (1933) gives a life table analysis of the times betweenmajor earthquake disasters, a technique that has been resurrected by several

more recent writers under the name of theory of durability An important

recent field of application has been the study of trains of nerve impulses in rophysiology In fact, the tools are available and relevant for any phenomenon

neu-in which the events occur along a time axis and the neu-intervals between the timepoints are important and meaningful quantities

Exercises and Complements to Section 1.1

1.1.1 A nonnegative random variable (r.v.) X with distribution function (d.f.) F has an increasing failure rate (abbreviated to IFR) if the conditional d.f.s

F x (u) = Pr {X ≤ x + u | X > x} = F (x + u) − F (x)

1− F (x) (u, x ≥ 0)

are increasing functions of x for every fixed u in 0 < u < ∞ It has a decreasing mean residual life (DMRL) if E(X − x | X > x) decreases with increasing x,

and it is new better than used in expectation (NBUE) if E(X − x | X > x) ≤

EX (all x > 0) Show that IFR implies DMRL, DMRL implies NBUE, and NBUE implies that var X ≤ (EX)2

[see Stoyan (1983, Section 1.6)]

1.1.2 Let X1, X2, be a sequence of independent identically distributed r.v.s with

d.f F ( ·) Then, for any fixed nonnegative integer n,

When F (u) = 1 −e −λu , G is the Gumbel d.f., while when F (u) = 1 −λu −α , G

is the Weibull d.f [In the forms indicated, these extreme-value distributionsinclude location and/or scale parameters; see e.g Johnson and Kotz (1970,

p 272).]

1.1.3 Let X1, X2, be as in the previous exercise with F (u) = 1 − e −λu Show

that Y ≡ max(X1, , X n) has the same distribution asn

j=1X j /j.

[Hint: Regard X1, , X n as lifetimes in a linear death process with death

rate λ, so that y is the time to extinction of the process Exercise 2.1.2 gives

more general properties.]

1.1.4 Suppose that the lifetimes of rulers are independent r.v.s with common d.f

F and that conditional on reaching age 21 years, a ruler has a son (with

lifetime d.f.s F ) every 2 years for up to six sons, with the eldest surviving son

succeeding him Conditional on there being a succession, what is the d.f ofthe age at succession and the expected time that successor reigns (assuming

a reign terminated by death from natural causes)?

What types of error would be involved in matching chronologies from aknowledge of the orders of two sets of rulers (see the reference to Newton’swork in the text)? How would such chronologies be matched in the light ofdevelopments in statistical techniques subsequent to Newton?

1.1.5 Investigate the integral equation for the stationary age distribution in a critical age-dependent branching process Using a suitable metric, evaluate thedifference between this stationary age distribution and the backward recur-rence time distribution of a stationary renewal process with the same lifetimedistribution as a function of the mean of the offspring distribution Note thatEuler worked on the age distribution in exponentially growing populations

The other basic approach to point process phenomena, and the only atic approach yet available in spaces of higher dimension, is to count thenumbers of events in intervals or regions of various types In this approach,the machinery of discrete distributions plays a central role Since in proba-bility theory discrete problems are usually easier to handle than continuousproblems, it might be thought that the development of general models for adiscrete distribution would precede those for a continuous distribution, but

system-in fact the reverse seems to be the case Although particular examples, such

as the Bernoulli distribution and the negative binomial distribution, occurred

at a very early stage in the discussion of games of chance, there seems to be

no discussion of discrete distributions as such until well into the nineteenthcentury

We may take as a starting point Poisson’s (1837) text, which included aderivation of the Poisson distribution by passage to the limit from the binomial(the claim that he was anticipated in this by de Moivre is a little exaggerated

in our view: it is true that de Moivre appends a limit result to the discussion

of a certain card problem, but it can hardly be said that the resulting formulawas considered by de Moivre as a distribution, which may be the key point).Even Poisson’s result does not seem to have been widely noted at the time,and it is not derived in a counting process context The first discussions

of counting problems known to us are by Seidel (1876) and Abb´e (1879),

who treated the occurrence of thunderstorms and the number of blood cells

in haemocytometer squares, respectively, and both apparently independently

of Poisson’s work Indeed, Poisson’s discovery of the distribution seems tohave been lost sight of until attention was drawn to it in Von Bortkiewicz’s

(1898) monograph Vas Gesetz der kleinen Zahlen, which includes a systematic

account of phenomena that fit the Poisson distribution, including, of course,the famous example of the number of deaths from horse kicks in the Prussianarmy

Lyon and Thoma (1881), on Abb´e’s data, and Student (1907) gave ther discussions of the blood cell problem, the latter paper being famous asone of the earliest applications of the chi-square goodness-of-fit test Shortlyafterward, the Poisson process arose simultaneously in two very importantcontexts Erlang (1909) derived the Poisson distribution for the number ofincoming calls to a telephone trunking system by supposing the numbers indisjoint intervals to be independent and considering the limit behaviour whenthe interval of observation is divided into an increasing number of equallysized subintervals This effectively reproduces the Poisson distribution as thelimit of the binomial, but Erlang was not aware of Poisson’s work at the time,although he corrected the omission in later papers Then, in 1910, Bateman,brought in as mathematical consultant by Rutherford and Geiger in connec-

fur-tion with their classical experiment on the counting of α particles, obtained

the Poisson probabilities as solutions to the family of differential equations

p 

n (t) = −λp n (t) + p n −1 (t) (n ≥ 1),

p 

0(t) = −λp0(t).

[Concerning the relation p0(t) = e −λt, Bateman (1910) commented that it

“has been known for some time (Whitworth’s Choice and Chance, 4th Ed.,

Proposition LI),” while Haight (1967) mentioned the result as a theorem ofBoltzmann (1868) and quoted the reference to Whitworth, who does not indi-cate the sources of his results; in a Gresham lecture reproduced in Whitworth(1897, p xxxiii), he wrote of Proposition LI as “a general theorem which Ipublished in 1886, which met with rather rough treatment at the hands of

a reviewer in The Academy.” Whitworth’s (1867) book evolved through five

editions It is easy to envisage repeated independent discovery of his sition LI.]

Propo-These equations represent a formulation in terms of a pure birth cess and the first step in the rapid development of the theory of birth anddeath processes during the next two decades, with notable early papers byMcKendrick (1914, 1926) and Yule (1924) This work preceded the generalformulation of birth and death processes as Markov processes (themselvesfirst studied by Markov more than a decade earlier) in the 1930s and is not ofimmediate concern, despite the close connection with point process problems

pro-A similar remark can be made about branching processes, studied first byBienaym´e (see Heyde and Seneta, 1977) and of course by Galton and Watson

(1874) There are close links with point processes, particularly in the eral case, but the early studies used special techniques that again lie a littleoutside the scope of our present discussion, and it was only from the 1940sonward that the links became important.

gen-Closer in line with our immediate interests is the work on alternatives to thePoisson distribution In many problems in ecology and elsewhere, it is foundthat the observed distribution of counts frequently shows a higher dispersion(i.e a higher variance for a given value of the mean) than can be accountedfor satisfactorily by the Poisson distribution, for which the variance/meanratio is identically unity The earliest and perhaps still the most widely used

alternative is the negative binomial distribution, which figures in early papers

by Student (1907), McKendrick (1914), and others A particularly importantpaper for the sequel was the study by Greenwood and Yule (1920) of accidentstatistics, which provided an important model for the negative binomial, and

in so doing sparked a controversy, still not entirely resolved, concerning theidentifiability of the model describing accident occurrence Since the accidentprocess is a kind of point process in time, and since shades of the same contro-versy will appear in our own models, we briefly paraphrase their derivation.Before doing so, however, it is convenient to summarize some of the machineryfor handling discrete distributions

The principal tool is the probability generating function (p.g.f.) defined for nonnegative integer-valued random variables X by the equation

P (z) =

∞

0

p n z n ,

where p n = Pr{X = n} It is worth mentioning that although generating

functions have been used in connection with difference equations at least sincethe time of Laplace, their application to this kind of problem in the 1920s and1930s was hailed as something of a technological breakthrough

In Chapter 5, relations between the p.g.f., factorial moments, and lants are discussed For the present, we content ourselves with the observationthat the negative binomial distribution can be characterized by the form ofits p.g.f.,

corresponding to values of the probabilities themselves,

1 + µ

n

1 Note that there is a lack of agreement on terminology Other authors, for example Johnson and Kotz (1969), would label this as a compound Poisson and would call the distribution

we treat below under that name a generalized Poisson The terminology we use is perhaps more common in texts on probability and stochastic processes; the alternative terminology

is more common in the statistical literature.

Greenwood and Yule derived this distribution as an example of what we

call a mixed Poisson1 distribution; that is, it can be obtained from a Poisson distribution p n = e−λ λ n /n! by treating the parameter λ as a random variable.

If, in particular, λ is assumed to have the gamma distribution

eλ(z −1) being the p.g.f of the Poisson distribution with parameter λ.

It is not difficult to verify that the mean and variance of this negative

bi-nomial distribution equal α/µ and (α/µ)(1 + µ −1), so that the variance/meanratio of the distribution equals 1 + µ −1 , exceeding by µ −1 the corresponding

ratio for a Poisson distribution Greenwood and Yule interpreted the variable

parameter λ of the underlying Poisson distribution as a measure of

individ-ual ‘accident proneness,’ which was then averaged over all individindivid-uals in thepopulation

The difficulty for the sequel is that, as was soon recognized, many othermodels also give rise to the negative binomial, and these may have quitecontradictory interpretations in regard to accidents L¨uders (1934) showed

that the same distribution could be derived as an example of a compound

Poisson distribution, meaning a random sum of independent random variables

in which the number of terms in the sum has a Poisson distribution If each

term is itself discrete and has a logarithmic distribution with p.g.f.

P (z) = log(1 + µ − z)

and if the number of terms has a Poisson distribution with parameter α,

then the resultant distribution has the identical p.g.f (1.2.1) for the negativebinomial (see Exercise 1.2.1) The interpretation here would be that all in-dividuals are identical but subject to accidents in batches Even before this,Eggenberger and P´olya (1923) and P´olya (1931) had introduced a whole fam-ily of distributions, for which they coined the term ‘contagious distributions’

to describe situations where the occurrence of a number of events enhancesthe probability of the occurrence of a further event, and had shown that thenegative binomial distribution could be obtained in this way If the mixedand compound models can be distinguished in principle by examining thejoint distributions of the number of accidents in nonoverlapping intervals of aperson’s life, Cane (1974, 1977) has shown that there is no way in which themixed Poisson and P´olya models can be distinguished from observations onindividual case histories, for they lead to identical conditional distributions(see Exercise 1.2.2)

Another important contribution in this field is the work of Neyman (1939),who introduced a further family of discrete distributions, derived from con-sideration of a cluster model Specifically, Neyman was concerned with distri-butions of beetle larvae in space, supposing these to have crawled some smalldistance from their initial locations in clusters of eggs Further analysis of thisproblem resulted in a series of papers, written by Neyman in collaborationwith E.L Scott and other writers, which treated many different statisticalquestions relating to clustering processes in ecology, astronomy, and othersubjects (see e.g Neyman and Scott, 1958).

Many of these questions can be treated most conveniently by the use ofgenerating functionals and moment densities, a theory that had been devel-oping simultaneously as a tool for describing the evolution of particle showersand related problems in theoretical physics The beginnings of such a generaltheory appear in the work of the French physicist Yvon (1935), but the maindevelopments relate to the post-war period, and we therefore defer a furtherdiscussion to the following section

Exercises and Complements to Section 1.2

1.2.1 Poisson mixture of logarithmic distributions is negative binomial Verify that

if X1, X2, are independent r.v.s with the logarithmic distribution whose

p.g.f is in (1.2.2), and if N , independent of X1, X2, , is a Poisson r.v with

mean α, then X1+· · · + X N has the negative binomial distribution in (1.2.1)

1.2.2 Nonidentifiability in a model for accident proneness Suppose that an ual has n accidents in the time interval (0, T ) at t1< t2< · · · < t n Evaluate

individ-the likelihood function for individ-these n times for individ-the two models:

(i) accidents occur at the epochs of a Poisson process at rate λ, where λ is

fixed for each individual but may vary between individuals;

(ii) conditional on having experienced j accidents in (0, t), an individual has probability (k + j)µ dt/(1 + µt) of an accident in (t, t + dt), independent

of the occurrence times of the j accidents in (0, t); each individual has probability kµ dt of an accident in (0, dt).

Show that the probabilities of n events in (0, T ) are Poisson and negative binomial, respectively, and deduce that the conditional likelihood, given n, is

the same for (i) and (ii) See Cane (1974) for discussion

1.2.3 The negative binomial distribution can also arise as the limit of the P´olya–

Eggenberger distribution defined for integers n and α, β > 0 by

Γ(α + β)n!Γ(β + n − k)

Γ(β)(n − k)!Γ(β + n + α) .

When β and n → ∞ with n/β → µ, a constant, and α fixed, show that {p k }

has the p.g.f in (1.2.1) [For further properties, see Johnson and Kotz (1969)and the papers cited in the text.]

1.2.4 Neyman’s Type A distribution (e.g Johnson and Kotz, 1969) has a p.g.f of

i α i = 1, λ i > 0, and µ > 0, and arises as a cluster model.

Give such a cluster model interpretation for the simplest case α i = 1 for

i = 1, α i = 0 otherwise, and general λ ≡ λ1and µ.

1.2.5 Suppose that a (large) population evolves according to a one-type Galton–Watson branching process in which the distribution of the number of children

has p.g.f P (z) Choose an individual at random in a particular generation.

Show that the distribution of the number of sibs (sisters, say) of this randomly

chosen individual has p.g.f P  (z)/P (1) and that this is the same as for thenumber of aunts, or great-aunts, of this individual

[Hint: Attempting to estimate the offspring distribution by using the observed

family size distribution, when based on sampling via the children, leads to

the distribution with p.g.f zP  (z)/P (1) and is an example of length-biased

sampling that underlies the waiting-time paradox referred to in Sections 3.2and 3.4 The p.g.f for the number of great-aunts is used in Chapter 11.]

The period during and following World War II saw an explosive growth intheory and applications of stochastic processes On the one hand, many newapplications were introduced and existing fields of application were extendedand deepened; on the other hand, there was also an attempt to unify the sub-ject by defining more clearly the basic theoretical concepts The monographs

by Feller (1950) and Bartlett (1955) (preceded by mimeographed lecture notesfrom 1947) played an important role in stressing common techniques and ex-ploring the mathematical similarities in different applications; both remainremarkably succinct and wide-ranging surveys

From such a busy scene it is difficult to pick out clearly marked lines ofdevelopment, and any selection of topics is bound to be influenced by personalpreferences Bearing such reservations in mind, we can attempt to followthrough some of the more important themes into the post-war period

On the queueing theory side, a paper of fundamental importance is nie Palm’s (1943) study of intensity fluctuations in traffic theory, a title thatembraces topics ranging from the foundation of a general theory of the in-put stream to the detailed analysis of particular telephone trunking systems.Three of his themes, in particular, were important for the future of pointprocesses The first is the systematic description of properties of a renewalprocess, as a first generalization of the Poisson process as input to a service

Con-system The notion of a regeneration point, a time instant at which the

sys-tem reverts to a specified state with the property that the future evolution isindependent of how the state was reached, has proved exceptionally fruitful

in many different applications In Palm’s terminology, the Poisson process

is characterized by the property that every instant is a regeneration point,whereas for a general renewal process only those instants at which a new in-terval is started form regeneration points Hence, he called a Poisson process

a process without aftereffects and a renewal process a process with limited aftereffects Another important idea was his realization that two types of dis-

tribution function are important in describing a stationary point process—thedistribution of the time to the next event from a fixed but arbitrary originand the distribution of the time to the next event from an arbitrary event

of the process The relations between the two sets of distributions are given

by a set of equations now commonly called the Palm–Khinchin equations,Palm himself having exhibited only the simplest special case A third im-portant contribution was his (incomplete) proof of the first limit theorem forpoint processes: namely, that superposition of a large number of independentsparse renewal processes leads to a Poisson process in the limit Finally, itmay be worth mentioning that it was in Palm’s paper that the term ‘pointprocesses’ (Punktprozesse) was first used as such—at least to the best of ourknowledge

All these ideas have led to important further development H Wold (1948,1949), also a Swedish mathematician, was one of the first to take up Palm’swork, studying processes with Markov-dependent intervals that, he suggested,would form the next most complex alternative to the renewal model Bartlett(1954) reviewed some of this early work Of the reworkings of Palm’s theory,however, the most influential was the monograph by Khinchin (1955), whichprovided a more complete and rigorous account of Palm’s work, notably ex-tended it in several directions, and had the very important effect of bringingthe subject to the attention of pure mathematicians Thus, Khinchin’s bookbecame the inspiration of much theoretical work, particularly in the SovietUnion and Eastern Europe Ryll-Nardzewski’s (1961) paper set out funda-mental properties of point processes and provided a new and more generalapproach to Palm probabilities Starting in the early 1960s, Matthes andco-workers developed many aspects concerned with infinitely divisible pointprocesses and related questions The book by Kerstan, Matthes and Mecke(1974) represented the culmination of the first decade of such work; extensiverevisions and new material were incorporated into the later editions in English(1978) (referred to as MKM in this book) and in Russian (1982)

In applications, these ideas have been useful not only in queueing theory

[for continuing development in this field, see the monographs of Franken et al.

(1981) and Br´emaud (1981)] but also in the study of level-crossing problems.Here the pioneering work was due to Rice (1944) and McFadden (1956, 1958).More rigorous treatments, using some of the Palm–Khinchin theory, weregiven by Leadbetter and other writers [see e.g Leadbetter (1972) and themonographs by Cram´er and Leadbetter (1967) and Leadbetter, Lindgren andRootzen (1983)]

On a personal note in respect of much of this work, it is appropriate toremark that Belyaev, Franken, Grigelionis, K¨onig, Matthes, and one of us,

among others, were affected by the lectures and personal influence of denko (see Vere-Jones, 1997), who was a student of Khinchin.

Gne-Meanwhile, there was also rapid development on the theoretical physicsfront The principal ideas here were the characteristic and generating func-tionals and product densities As early as 1935, Kolmogorov suggested theuse of the characteristic functional

of specific population models

Of more immediate relevance to point processes is the related concept of a

probability generating functional (p.g.fl.) defined by

G[h] = E 

i h(x i)



= E

exp

 

log h(x) N (dx)



,

where h(x) is a suitable test function and the x iare the points at which

popu-lation members are located, that is, the atoms of the counting measures N ( ·).

The p.g.fl is the natural extension of the p.g.f., and, like the p.g.f., it has anexpansion, when the total population is finite, in terms of the probabilities ofthe number of particles in the population and the probability densities of theirlocations There is also an expansion, analogous to the expansion of the p.g.f

in terms of factorial moments, in terms of certain factorial moment density

functions, or product densities as they are commonly called in the physical

literature Following the early work of Yvon noted at the end of Section1.2, the p.g.fl and product densities were used by Bogoliubov (1946), whileproperties of product densities were further explored in important papers byBhabha (1950) and Ramakrishnan (1950) Ramakrishnan, in particular, gaveformulae expressing the moments of the number of particles in a given set interms of the product densities and Stirling numbers Later, these ideas wereconsiderably extended by Ramakrishnan, Janossy, Srinivasan, and others; anextensive literature exists on their application to cosmic ray showers summa-rized in the monographs by Janossy (1948) and Srinivasan (1969, 1974)

This brings us to another key point in the mathematical theory of pointprocesses, namely the fundamental paper by Moyal (1962a) Drawing princi-pally on the physical and ecological contexts, Moyal for the first time set outclearly the mathematical constructs needed for a theory of point processes on

a general state space, clarifying the relations between such quantities as theproduct densities, finite-dimensional distributions, and probability generatingfunctionals and pointing out a number of important applications Indepen-dently, Harris (1963) set out similar ideas in his monograph on branchingprocesses, subsequently (Harris, 1968, 1971) contributing important ideas tothe general theory of point processes and the more complex subject of inter-acting particle systems

In principle, the same techniques are applicable to other contexts wherepopulation models are important, but in practice the discussions in such con-

texts have tended to use more elementary, ad hoc tools In forestry, for

exam-ple, a key problem is the assessment of the number of diseased or other specialkinds of trees in a given region Since a complete count may be physicallyvery difficult to carry out and expensive, emphasis has been on statistical sam-pling techniques, particularly of transects (line segments drawn through theregion) and nearest-neighbour distances Mat´ern’s (1960) monograph broughttogether many ideas, models, and statistical techniques of importance in suchfields and includes an account of point process aspects Ripley’s (1981) mono-graph covers some more recent developments

On the statistical side, Cox’s (1955) paper contained seeds leading to thetreatment of many statistical questions concerning data generated by pointprocesses and discussing various models, including the important class of dou-bly stochastic Poisson processes A further range of techniques was introduced

by Bartlett (1963), who showed how to adapt methods of time series analysis

to a point process context and brought together a variety of different models.This work was extended to processes in higher dimensions in a second paper(Bartlett, 1964) Lewis (1964a) used similar techniques to discuss the instants

of failure of a computer The subsequent monograph by Cox and Lewis (1966)was a further important development that, perhaps for the first time, showedclearly the wide range of applications of point processes as well as extendingmany of the probabilistic and statistical aspects of such processes

In the 1970s, perhaps the most important development was the rapidgrowth of interest in point processes in communications engineering (see e.g.Snyder, 1975) It is a remarkable fact that in nature, for example in nervesystems, the transfer of information is more often effected by pulse signalsthan by continuous signals This fact seems to be associated with the highsignal/noise ratios that it is possible to achieve by these means; for the samereason, pulse techniques are becoming increasingly important in communica-tion applications For such processes, just as for continuous processes, it ismeaningful to pose questions concerning the prediction, interpolation, and es-timation of signals, and the detection of signals against background noise (inthis context, of random pulses) Since the signals are intrinsically nonnega-

tive, the distributions cannot be Gaussian, so linear models are not in generalappropriate Thus, the development of a suitable theory for point processes

is closely linked to the development of nonlinear techniques in other branches

of stochastic process theory As in the applications to processes of diffusiontype, martingale methods provide a powerful tool in the discussion of theseproblems, yielding, for example, structural information about the process andits likelihood function as well as more technical convergence results Amongstother books, developments in this area were surveyed in Liptser and Shiryayev(1974; English translation 1977, 1978; 2nd ed 2000), Br´emaud (1981), andJacobsen (1982)

The last quarter-century has seen both the emergence of new fields of plications and the consolidation of older ones Here we shall attempt no morethan a brief indication of major directions, with references to texts that can

ap-be consulted for more substantive treatments

Spatial point processes, or spatial point patterns as they are often called,

have become a burgeoning subject in their own right The many fields ofapplication include environmental studies, ecology, geography, astrophysics,fisheries and forestry, as well as substantially new topics such as image pro-cessing and spatial epidemic theory Ripley (1981) and Diggle (1983) discussboth models and statistical procedures, while Cressie (1991) gives a broadoverview with the emphasis on applications in biology and ecology Imageprocessing is discussed in the now classical work of Serra (1982) Theoreticalaspects of spatial point patterns link closely with the fields of stereology and

stochastic geometry, stemming from the seminal work of Roger Miles and,

particularly, Rollo Davidson (see Harding and Kendall, 1974) and surveyed

in Stoyan, Kendall and Mecke (1987, 2nd ed 1995) and Stoyan and Stoyan(1994) There are also close links with the newly developing subject of randomset theory; see Math´eron (1975) and Molchanov (1997) The broad-ranging

set of papers in Barndorff-Nielsen et al (1998) covers many of these

applica-tions and associated theory

Time, space–time, and marked space–time point processes have ued to receive considerable attention As well as in the earlier applications

contin-to queueing theory, reliability, and electrical engineering, they have foundimportant uses in geophysics, neurophysiology, cardiology, finance, and eco-nomics Applications in queueing theory and reliability were developed in the1980s by Br´emaud (1981) and Franken et al (1981) Baccelli and Br´emaud(1994) contains a more recent account Second-order methods for the statis-tical analysis of such data, including spectral theory, are outlined in the nowclassic text of Cox and Lewis (1966) and in Brillinger (1975b) Snyder andMiller (1991) describe some of the more recent applications in medical fields.Extreme-value ideas in finance are discussed, from a rather different point

of view than in Leadbetter et al (1983) and Resnick (1987), in Embrechts

et al (1997) Prediction methods for point processes have assumed growing

importance in seismological applications, in which context they are reviewed

in Vere-Jones (1995)

Survival analysis has emerged as another closely related major topic, with

applications in epidemiology, medicine, mortality, quality control, reliability,and other fields Here the study of a single point process is usually replaced bythe study of many individual processes, sometimes with only a small number

of events in each, evolving simultaneously Starting points include the early

papers of Cox (1972b) and Aalen (1975) Andersen et al (1993) give a

ma-jor survey of modelling and inference problems in this field; their treatmentincludes an excellent introduction to point process concepts in general, em-phasizing martingale concepts for inference, and the use of product-integralformulae

The growing range of applications has led to an upsurge of interest in ence problems for point process models Many of the texts referred to abovedevote a substantial part of their discussion to the practical implementation

infer-of inference procedures General principles infer-of inference for point processesare treated in the text by Liptser and Shiryayev already mentioned and inKutoyants (1980, 1984), Karr (1986, 2nd ed 1991), and Kutoyants (1998).Theoretical aspects have also continued to flourish, particularly in the con-nections with statistical mechanics and stochastic geometry Recent texts onbasic theory include Kingman’s (1993) beautiful discussion of the Poisson pro-cess and Last and Brandt’s (1995) exposition of marked point processes Thereare close connections between point processes and infinite particle systems(Liggett, 1999), while Georgii (1988) outlines ideas related to spatial processesand phase changes Branching processes in higher-dimensional spaces exhibit

many remarkable characteristics, some of which are outlined in Dawson et al.

(2000) Very recently, Coram and Diaconis (2002), exploiting Diaconis andEvans (2000, 2001), have studied similarities between finite point processes of

n points on the unit circle constructed from the eigenvalues of random unitary matrices from the unitary group U n , and blocks of n successive zeros of the Riemann zeta function, where n depends on the distance from the real axis

of the block of zeros