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Lectures on the Poisson Process
Lectures on the Poisson Process
Gunter Last, Mathew Penrose
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The Poisson process, a core object in modern probability, enjoys a richer theory than is sometimes appreciated. This volume develops the theory in the setting of a general abstract measure space, establishing basic results and properties as well as certain advanced topics in the stochastic analysis of the Poisson process. Also discussed are applications and related topics in stochastic geometry, including stationary point processes, the Boolean model, the Gilbert graph, stable allocations, and hyperplane processes. Comprehensive, rigorous, and selfcontained, this text is ideal for graduate courses or for selfstudy, with a substantial number of exercises for each chapter. Mathematical prerequisites, mainly a sound knowledge of measuretheoretic probability, are kept in the background, but are reviewed comprehensively in the appendix. The authors are wellknown researchers in probability theory; especially stochastic geometry. Their approach is informed both by their research and by their extensive experience in teaching at undergraduate and graduate levels.
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2018
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Cambridge University Press
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english
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315
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9781316104477
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Institute of Mathematical Statistics Textbooks 7
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‘An understanding of the remarkable properties of the Poisson process is essential for anyone interested in the mathematical theory of probability or in its many ﬁelds of application. This book is a lucid and thorough account, rigorous but not pedantic, and accessible to any reader familiar with modern mathematics at ﬁrstdegree level. Its publication is most welcome.’ — J. F. C. Kingman, University of Bristol ‘I have always considered the Poisson process to be a cornerstone of applied probability. This excellent book demonstrates that it is a whole world in and of itself. The text is exciting and indispensable to anyone who works in this ﬁeld.’ — Dietrich Stoyan, TU Bergakademie Freiberg ‘Last and Penrose’s Lectures on the Poisson Process constitutes a splendid addition to the monograph literature on point processes. While emphasising the Poisson and related processes, their mathematical approach also covers the basic theory of random measures and various applications, especially to stochastic geometry. They assume a sound grounding in measuretheoretic probability, which is well summarised in two appendices (on measure and probability theory). Abundant exercises conclude each of the twentytwo “lectures” which include examples illustrating their “course” material. It is a ﬁrstclass complement to John Kingman’s essay on the Poisson process.’ — Daryl Daley, University of Melbourne ‘Pick n points uniformly and independently in a cube of volume n in Euclidean space. The limit of these random conﬁgurations as n → ∞ is the Poisson process. This book, written by two of the foremost experts on point processes, gives a masterful overview of the Poisson process and some of its relatives. Classical tenets of the theory, like thinning properties and Campbell’s formula, are followed by modern developments, such as Liggett’s extra heads theorem, Fock space, permanental processes and the Boolean model. Numerous exercises throughout the book challenge readers and bring them to the edge of current theory.’ — Yuval Peres; , Principal Researcher, Microsoft Research, and Foreign Associate, National Academy of Sciences Lectures on the Poisson Process The Poisson process, a core object in modern probability, enjoys a richer theory than is sometimes appreciated. This volume develops the theory in the setting of a general abstract measure space, establishing basic results and properties as well as certain advanced topics in the stochastic analysis of the Poisson process. Also discussed are applications and related topics in stochastic geometry, including stationary point processes, the Boolean model, the Gilbert graph, stable allocations and hyperplane processes. Comprehensive, rigorous, and selfcontained, this text is ideal for graduate courses or for selfstudy, with a substantial number of exercises for each chapter. Mathematical prerequisites, mainly a sound knowledge of measuretheoretic probability, are kept in the background, but are reviewed comprehensively in an appendix. The authors are wellknown researchers in probability theory, especially stochastic geometry. Their approach is informed both by their research and by their extensive experience in teaching at undergraduate and graduate levels. G Ü N T E R L A S T is Professor of Stochastics at the Karlsruhe Institute of Technology. He is a distinguished probabilist with particular expertise in stochastic geometry, point processes and random measures. He has coauthored a research monograph on marked point processes on the line as well as two textbooks on general mathematics. He has given many invited talks on his research worldwide. M AT H E W P E N R O S E is Professor of Probability at the University of Bath. He is an internationally leading researcher in stochastic geometry and applied probability and is the author of the inﬂuential monograph Random Geometric Graphs. He received the Friedrich Wilhelm Bessel Research Award from the Humboldt Foundation in 2008, and has held visiting positions as guest lecturer in New Delhi, Karlsruhe, San Diego, Birmingham and Lille. I N S T I T U T E O F M AT H E M AT I C A L S TAT I S T I C S TEXTBOOKS Editorial Board D. R. Cox (University of Oxford) B. Hambly (University of Oxford) S. Holmes (Stanford University) J. Wellner (University of Washington) IMS Textbooks give introductory accounts of topics of current concern suitable for advanced courses at master’s level, for doctoral students and for individual study. They are typically shorter than a fully developed textbook, often arising from material created for a topical course. Lengths of 100–290 pages are envisaged. The books typically contain exercises. Other Books in the Series 1. 2. 3. 4. 5. Probability on Graphs, by Geoffrey Grimmett Stochastic Networks, by Frank Kelly and Elena Yudovina Bayesian Filtering and Smoothing, by Simo Särkkä The Surprising Mathematics of Longest Increasing Subsequences, by Dan Romik Noise Sensitivity of Boolean Functions and Percolation, by Christophe Garban and Jeffrey E. Steif 6. Core Statistics, by Simon N. Wood 7. Lectures on the Poisson Process, by Günter Last and Mathew Penrose Lectures on the Poisson Process GÜNTER LAST Karlsruhe Institute of Technology M AT H E W P E N RO S E University of Bath University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107088016 DOI: 10.1017/9781316104477 c Günter Last and Mathew Penrose 2018 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2018 Printed in the United States of America by Sheridan Books, Inc. A catalogue record for this publication is available from the British Library. Library of Congress CataloginginPublication Data Names: Last, Günter, author.  Penrose, Mathew, author. Title: Lectures on the Poisson process / Günter Last, Karlsruhe Institute of Technology, Mathew Penrose, University of Bath. Description: Cambridge : Cambridge University Press, 2018.  Series: Institute of Mathematical Statistics textbooks  Includes bibliographical references and index. Identiﬁers: LCCN 2017027687  ISBN 9781107088016 Subjects: LCSH: Poisson processes.  Stochastic processes.  Probabilities. Classiﬁcation: LCC QA274.42 .L36 2018  DDC 519.2/4–dc23 LC record available at https://lccn.loc.gov/2017027687 ISBN 9781107088016 Hardback ISBN 9781107458437 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or thirdparty internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. To Our Families Contents page xv xix Preface List of Symbols 1 1 3 4 5 7 1 1.1 1.2 1.3 1.4 1.5 Poisson and Other Discrete Distributions The Poisson Distribution Relationships Between Poisson and Binomial Distributions The Poisson Limit Theorem The Negative Binomial Distribution Exercises 2 2.1 2.2 2.3 2.4 2.5 Point Processes Fundamentals Campbell’s Formula Distribution of a Point Process Point Processes on Metric Spaces Exercises 9 9 12 14 16 18 3 3.1 3.2 3.3 3.4 Poisson Processes Deﬁnition of the Poisson Process Existence of Poisson Processes Laplace Functional of the Poisson Process Exercises 19 19 20 23 24 4 4.1 4.2 4.3 4.4 4.5 The Mecke Equation and Factorial Measures The Mecke Equation Factorial Measures and the Multivariate Mecke Equation Janossy Measures Factorial Moment Measures Exercises 26 26 28 32 34 36 ix x Contents 5 5.1 5.2 5.3 5.4 Mappings, Markings and Thinnings Mappings and Restrictions The Marking Theorem Thinnings Exercises 38 38 39 42 44 6 6.1 6.2 6.3 6.4 6.5 6.6 Characterisations of the Poisson Process Borel Spaces Simple Point Processes Rényi’s Theorem Completely Orthogonal Point Processes Turning Distributional into Almost Sure Identities Exercises 46 46 49 50 52 54 56 7 7.1 7.2 7.3 7.4 7.5 Poisson Processes on the Real Line The Interval Theorem Marked Poisson Processes Record Processes Polar Representation of Homogeneous Poisson Processes Exercises 58 58 61 63 65 66 8 8.1 8.2 8.3 8.4 8.5 8.6 Stationary Point Processes Stationarity The Pair Correlation Function Local Properties Ergodicity A Spatial Ergodic Theorem Exercises 69 69 71 74 75 77 80 9 9.1 9.2 9.3 9.4 9.5 The Palm Distribution Deﬁnition and Basic Properties The Mecke–Slivnyak Theorem Local Interpretation of Palm Distributions Voronoi Tessellations and the Inversion Formula Exercises 82 82 84 85 87 89 10 Extra Heads and Balanced Allocations 10.1 The Extra Head Problem 10.2 The PointOptimal Gale–Shapley Algorithm 10.3 Existence of Balanced Allocations 10.4 Allocations with Large Appetite 10.5 The Modiﬁed Palm Distribution 10.6 Exercises 92 92 95 97 99 101 101 Contents xi 11 Stable Allocations 11.1 Stability 11.2 The SiteOptimal Gale–Shapley Allocation 11.3 Optimality of the Gale–Shapley Algorithms 11.4 Uniqueness of Stable Allocations 11.5 Moment Properties 11.6 Exercises 103 103 104 104 107 108 109 12 Poisson Integrals 12.1 The Wiener–Itô Integral 12.2 Higher Order Wiener–Itô Integrals 12.3 Poisson UStatistics 12.4 Poisson Hyperplane Processes 12.5 Exercises 111 111 114 118 122 124 13 Random Measures and Cox Processes 13.1 Random Measures 13.2 Cox Processes 13.3 The Mecke Equation for Cox Processes 13.4 Cox Processes on Metric Spaces 13.5 Exercises 127 127 129 131 132 133 14 Permanental Processes 14.1 Deﬁnition and Uniqueness 14.2 The Stationary Case 14.3 Moments of Gaussian Random Variables 14.4 Construction of Permanental Processes 14.5 Janossy Measures of Permanental Cox Processes 14.6 OneDimensional Marginals of Permanental Cox Processes 14.7 Exercises 136 136 138 139 141 145 147 151 15 Compound Poisson Processes 15.1 Deﬁnition and Basic Properties 15.2 Moments of Symmetric Compound Poisson Processes 15.3 Poisson Representation of Completely Random Measures 15.4 Compound Poisson Integrals 15.5 Exercises 153 153 157 158 161 163 16 The Boolean Model and the Gilbert Graph 16.1 Capacity Functional 16.2 Volume Fraction and Covering Property 16.3 Contact Distribution Functions 16.4 The Gilbert Graph 166 166 168 170 171 xii 16.5 16.6 Contents The Point Process of Isolated Nodes Exercises 176 177 17 The Boolean Model with General Grains 17.1 Capacity Functional 17.2 Spherical Contact Distribution Function and Covariance 17.3 Identiﬁability of Intensity and Grain Distribution 17.4 Exercises 179 179 182 183 185 18 Fock Space and Chaos Expansion 18.1 Diﬀerence Operators 18.2 Fock Space Representation 18.3 The Poincaré Inequality 18.4 Chaos Expansion 18.5 Exercises 187 187 189 193 194 195 19 Perturbation Analysis 19.1 A Perturbation Formula 19.2 Power Series Representation 19.3 Additive Functions of the Boolean Model 19.4 Surface Density of the Boolean Model 19.5 Mean Euler Characteristic of a Planar Boolean Model 19.6 Exercises 197 197 200 203 206 207 208 20 Covariance Identities 20.1 Mehler’s Formula 20.2 Two Covariance Identities 20.3 The Harris–FKG Inequality 20.4 Exercises 211 211 214 217 217 21 Normal Approximation 21.1 Stein’s Method 21.2 Normal Approximation via Diﬀerence Operators 21.3 Normal Approximation of Linear Functionals 21.4 Exercises 219 219 221 225 226 22 Normal Approximation in the Boolean Model 22.1 Normal Approximation of the Volume 22.2 Normal Approximation of Additive Functionals 22.3 Central Limit Theorems 22.4 Exercises 227 227 230 235 237 Contents xiii Appendix A Some Measure Theory A.1 General Measure Theory A.2 Metric Spaces A.3 Hausdorﬀ Measures and Additive Functionals A.4 Measures on the Real HalfLine A.5 Absolutely Continuous Functions 239 239 250 252 257 259 Appendix B Some Probability Theory B.1 Fundamentals B.2 Mean Ergodic Theorem B.3 The Central Limit Theorem and Stein’s Equation B.4 Conditional Expectations B.5 Gaussian Random Fields 261 261 264 266 268 269 Appendix C Historical Notes 272 References Index 281 289 Preface The Poisson process generates point patterns in a purely random manner. It plays a fundamental role in probability theory and its applications, and enjoys a rich and beautiful theory. While many of the applications involve point processes on the line, or more generally in Euclidean space, many others do not. Fortunately, one can develop much of the theory in the abstract setting of a general measurable space. We have prepared the present volume so as to provide a modern textbook on the general Poisson process. Despite its importance, there are not many monographs or graduate texts with the Poisson process as their main point of focus, for example by comparison with the topic of Brownian motion. This is probably due to a viewpoint that the theory of Poisson processes on its own is too insubstantial to merit such a treatment. Such a viewpoint now seems out of date, especially in view of recent developments in the stochastic analysis of the Poisson process. We also extend our remit to topics in stochastic geometry, which is concerned with mathematical models for random geometric structures [4, 5, 23, 45, 123, 126, 147]. The Poisson process is fundamental to stochastic geometry, and the applications areas discussed in this book lie largely in this direction, reﬂecting the taste and expertise of the authors. In particular, we discuss Voronoi tessellations, stable allocations, hyperplane processes, the Boolean model and the Gilbert graph. Besides stochastic geometry, there are many other ﬁelds of application of the Poisson process. These include Lévy processes [10, 83], Brownian excursion theory [140], queueing networks [6, 149], and Poisson limits in extreme value theory [139]. Although we do not cover these topics here, we hope nevertheless that this book will be a useful resource for people working in these and related areas. This book is intended to be a basis for graduate courses or seminars on the Poisson process. It might also serve as an introduction to point process theory. Each chapter is supposed to cover material that can be presented xv xvi Preface (at least in principle) in a single lecture. In practice, it may not always be possible to get through an entire chapter in one lecture; however, in most chapters the most essential material is presented in the early part of the chapter, and the later part could feasibly be left as background reading if necessary. While it is recommended to read the earlier chapters in a linear order at least up to Chapter 5, there is some scope for the reader to pick and choose from the later chapters. For example, a reader more interested in stochastic geometry could look at Chapters 8–11 and 16–17. A reader wishing to focus on the general abstract theory of Poisson processes could look at Chapters 6, 7, 12, 13 and 18–21. A reader wishing initially to take on slightly easier material could look at Chapters 7–9, 13 and 15–17. The book divides loosely into three parts. In the ﬁrst part we develop basic results on the Poisson process in the general setting. In the second part we introduce models and results of stochastic geometry, most but not all of which are based on the Poisson process, and which are most naturally developed in the Euclidean setting. Chapters 8, 9, 10, 16, 17 and 22 are devoted exclusively to stochastic geometry while other chapters use stochastic geometry models for illustrating the theory. In the third part we return to the general setting and describe more advanced results on the stochastic analysis of the Poisson process. Our treatment requires a sound knowledge of measuretheoretic probability theory. However, speciﬁc knowledge of stochastic processes is not assumed. Since the focus is always on the probabilistic structure, technical issues of measure theory are kept in the background, whenever possible. Some basic facts from measure and probability theory are collected in the appendices. When treating a classical and central subject of probability theory, a certain overlap with other books is inevitable. Much of the material of the earlier chapters, for instance, can also be found (in a slightly more restricted form) in the highly recommended book [75] by J.F.C. Kingman. Further results on Poisson processes, as well as on general random measures and point processes, are presented in the monographs [6, 23, 27, 53, 62, 63, 69, 88, 107, 134, 139]. The recent monograph Kallenberg [65] provides an excellent systematic account of the modern theory of random measures. Comments on the early history of the Poisson process, on the history of the main results presented in this book and on the literature are given in Appendix C. In preparing this manuscript we have beneﬁted from comments on earlier versions from Daryl Daley, Fabian Gieringer, Christian Hirsch, Daniel Hug, Olav Kallenberg, Paul Keeler, Martin Möhle, Franz Nestmann, Jim Preface xvii Pitman, Matthias Schulte, Tomasz Rolski, Dietrich Stoyan, Christoph Thäle, Hermann Thorisson and Hans Zessin, for which we are most grateful. Thanks are due to Franz Nestmann for producing the ﬁgures. We also wish to thank Olav Kallenberg for making available to us an early version of his monograph [65]. Günter Last Mathew Penrose Symbols Z = {0, 1, −1, 2, −2, . . .} N = {1, 2, 3, 4, . . .} N0 = {0, 1, 2, . . .} set of integers set of positive integers set of nonnegative integers N = N ∪ {∞} extended set of positive integers N0 = N0 ∪ {∞} R = (−∞, ∞), R+ = [0, ∞) extended set of nonnegative integers real line (resp. nonnegative real halfline) R = R ∪ {−∞, ∞} extended real line R+ = R+ ∪ {∞} = [0, ∞] R(X), R+ (X) extended halfline Rvalued (resp. R+ valued) measurable functions on X R(X), R+ (X) Rvalued (resp. R+ valued) measurable functions on X + − u ,u positive and negative part of an Rvalued function u a ∧ b, a ∨ b 1{·} minimum (resp. maximum) of a, b ∈ R indicator function a⊕ := 1{a 0}a−1 card A = A [n] Σn Πn , Π∗n (n)k = n · · · (n − k + 1) δx N<∞ (X) ≡ N<∞ N(X) ≡ N Nl (X), N s (X) Nls (X) := Nl (X) ∩ N s (X) x∈μ νB generalised inverse of a ∈ R number of elements of a set A {1, . . . , n} group of permutations of [n] set of all partitions (resp. subpartitions) of [n] descending factorial Dirac measure at the point x set of all ﬁnite counting measures on X set of all countable sums of measures from N<∞ set of all locally ﬁnite (resp. simple) measures in N(X) set of all locally ﬁnite and simple measures in N(X) short for μ{x} = μ({x}) > 0, μ ∈ N restriction of a measure ν to a measurable set B xix List of Symbols xx B(X) Xb Borel σﬁeld on a metric space X bounded Borel subsets of a metric space X Rd Euclidean space of dimension d ∈ N Bd := B(Rd ) Borel σﬁeld on Rd λd Lebesgue measure on (Rd , Bd ) · Euclidean norm on Rd ·, · Euclidean scalar product on Rd C ,C compact (resp. nonempty compact) subsets of Rd K d , K (d) compact (resp. nonempty compact) convex subsets of Rd Rd convex ring in Rd (ﬁnite unions of convex sets) K + x, K − x translation of K ⊂ Rd by x (resp. −x) K⊕L V0 , . . . , Vd φi = Vi (K) Q(dK) Minkowski sum of K, L ⊂ Rd intrinsic volumes ith mean intrinsic volume of a typical grain (d) d B(x, r) closed ball with centre x and radius r ≥ 0 κd = λd (Bd ) volume of the unit ball in Rd < strict lexicographical order on Rd l(B) (Ω, F , P) E[X] Var[X] Cov[X, Y] Lη lexicographic minimum of a nonempty ﬁnite set B ⊂ Rd probability space expectation of a random variable X variance of a random variable X covariance between random variables X and Y Laplace functional of a random measure η d d =, → equality (resp. convergence) in distribution 1 Poisson and Other Discrete Distributions The Poisson distribution arises as a limit of the binomial distribution. This chapter contains a brief discussion of some of its fundamental properties as well as the Poisson limit theorem for null arrays of integervalued random variables. The chapter also discusses the binomial and negative binomial distributions. 1.1 The Poisson Distribution A random variable X is said to have a binomial distribution Bi(n, p) with parameters n ∈ N0 := {0, 1, 2, . . .} and p ∈ [0, 1] if n k P(X = k) = Bi(n, p; k) := p (1 − p)n−k , k = 0, . . . , n, (1.1) k where 00 := 1. In the case n = 1 this is the Bernoulli distribution with parameter p. If X1 , . . . , Xn are independent random variables with such a Bernoulli distribution, then their sum has a binomial distribution, that is d X1 + · · · + Xn = X, (1.2) d where X has the distribution Bi(n, p) and where = denotes equality in distribution. It follows that the expectation and variance of X are given by E[X] = np, Var[X] = np(1 − p). (1.3) A random variable X is said to have a Poisson distribution Po(γ) with parameter γ ≥ 0 if P(X = k) = Po(γ; k) := γk −γ e , k! k ∈ N0 . (1.4) If γ = 0, then P(X = 0) = 1, since we take 00 = 1. Also we allow γ = ∞; in this case we put P(X = ∞) = 1 so Po(∞; k) = 0 for k ∈ N0 . The Poisson distribution arises as a limit of binomial distributions as 1 Poisson and Other Discrete Distributions 2 follows. Let pn ∈ [0, 1], n ∈ N, be a sequence satisfying npn → γ as n → ∞, with γ ∈ (0, ∞). Then, for k ∈ {0, . . . , n}, npn n γk (npn )k (n)k n k · k · (1 − pn )−k · 1 − → e−γ , (1.5) pn (1 − pn )n−k = k! n n k! k as n → ∞, where (n)k := n(n − 1) · · · (n − k + 1) (1.6) is the kth descending factorial (of n) with (n)0 interpreted as 1. Suppose X is a Poisson random variable with ﬁnite parameter γ. Then its expectation is given by E[X] = e−γ ∞ ∞ γk−1 γk = γ. k = e−γ γ k! (k − 1)! k=0 k=1 (1.7) The probability generating function of X (or of Po(γ)) is given by Es X −γ =e ∞ γk k=0 k! s =e k −γ ∞ (γs)k k=0 k! = eγ(s−1) , s ∈ [0, 1]. It follows that the Laplace transform of X (or of Po(γ)) is given by E e−tX = exp[−γ(1 − e−t )], t ≥ 0. (1.8) (1.9) Formula (1.8) is valid for each s ∈ R and (1.9) is valid for each t ∈ R. A calculation similar to (1.8) shows that the factorial moments of X are given by E[(X)k ] = γk , k ∈ N0 , (1.10) where (0)0 := 1 and (0)k := 0 for k ≥ 1. Equation (1.10) implies that Var[X] = E[X 2 ] − E[X]2 = E[(X)2 ] + E[X] − E[X]2 = γ. (1.11) We continue with a characterisation of the Poisson distribution. Proposition 1.1 An N0 valued random variable X has distribution Po(γ) if and only if, for every function f : N0 → R+ , we have E[X f (X)] = γ E[ f (X + 1)]. (1.12) Proof By a similar calculation to (1.7) and (1.8) we obtain for any function f : N0 → R+ that (1.12) holds. Conversely, if (1.12) holds for all such functions f , then we can make the particular choice f := 1{k} for k ∈ N, to obtain the recursion k P(X = k) = γ P(X = k − 1). 1.2 Relationships Between Poisson and Binomial Distributions 3 This recursion has (1.4) as its only (probability) solution, so the result follows. 1.2 Relationships Between Poisson and Binomial Distributions The next result says that if X and Y are independent Poisson random variables, then X + Y is also Poisson and the conditional distribution of X given X + Y is binomial: Proposition 1.2 Let X and Y be independent with distributions Po(γ) and Po(δ), respectively, with 0 < γ+δ < ∞. Then X+Y has distribution Po(γ+δ) and P(X = k  X + Y = n) = Bi(n, γ/(γ + δ); k), Proof n ∈ N0 , k = 0, . . . , n. For n ∈ N0 and k ∈ {0, . . . , n}, δn−k −δ γk e P(X = k, X + Y = n) = P(X = k, Y = n − k) = e−γ k! (n − k)! (γ + δ)n n γ k δ n−k = e−(γ+δ) n! k γ+δ γ+δ = Po(γ + δ; n) Bi(n, γ/(γ + δ); k), and the assertions follow. Let Z be an N0 valued random variable and let Z1 , Z2 , . . . be a sequence of independent random variables that have a Bernoulli distribution with parameter p ∈ [0, 1]. If Z and (Zn )n≥1 are independent, then the random variable X := Z Zj (1.13) j=1 is called a pthinning of Z, where we set X := 0 if Z = 0. This means that the conditional distribution of X given Z = n is binomial with parameters n and p. The following partial converse of Proposition 1.2 is a noteworthy property of the Poisson distribution. Proposition 1.3 Let p ∈ [0, 1]. Let Z have a Poisson distribution with parameter γ ≥ 0 and let X be a pthinning of Z. Then X and Z − X are independent and Poisson distributed with parameters pγ and (1 − p)γ, respectively. Poisson and Other Discrete Distributions 4 Proof that We may assume that γ > 0. The result follows once we have shown P(X = m, Z − X = n) = Po(pγ; m) Po((1 − p)γ; n), m, n ∈ N0 . (1.14) Since the conditional distribution of X given Z = m + n is binomial with parameters m + n and p, we have P(X = m, Z − X = n) = P(Z = m + n) P(X = m  Z = m + n) −γ m+n m+n m e γ p (1 − p)n = (m + n)! m m m n n p γ −pγ (1 − p) γ e e−(1−p)γ , = m! n! and (1.14) follows. 1.3 The Poisson Limit Theorem The next result generalises (1.5) to sums of Bernoulli variables with unequal parameters, among other things. Proposition 1.4 Suppose for n ∈ N that mn ∈ N and Xn,1 , . . . , Xn,mn are independent random variables taking values in N0 . Let pn,i := P(Xn,i ≥ 1) and assume that lim max pn,i = 0. (1.15) n→∞ 1≤i≤mn Assume further that λn := mn i=1 lim n→∞ Let Xn := mn i=1 pn,i → γ as n → ∞, where γ > 0, and that mn P(Xn,i ≥ 2) = 0. (1.16) i=1 Xn,i . Then for k ∈ N0 we have lim P(Xn = k) = Po(γ; k). (1.17) n→∞ := 1{Xn,i ≥ 1} = min{Xn,i , 1} and Xn := Proof Let Xn,i Xn,i Xn,i if and only if Xn,i ≥ 2, we have P(Xn Xn ) ≤ mn mn i=1 Xn,i . Since P(Xn,i ≥ 2). i=1 By assumption (1.16) we can assume without restriction of generality that 1.4 The Negative Binomial Distribution 5 Xn,i = Xn,i for all n ∈ N and i ∈ {1, . . . , mn }. Moreover it is no loss of generality to assume for each (n, i) that pn,i < 1. We then have P(Xn = k) = pn,i1 pn,i2 · · · pn,ik 1≤i1 <i2 <···<ik ≤mn Let μn := max1≤i≤mn pn,i . Since mn log mn j=1 mn j=1 (1 − pn, j ) (1 − pn,i1 ) · · · (1 − pn,ik ) p2n, j ≤ λn μn → 0 as n → ∞, we have mn (1 − pn, j ) = (−pn, j + O(p2n, j )) → −γ as n → ∞, j=1 . (1.18) (1.19) j=1 where the function O(·) satisﬁes lim supr→0 r−1 O(r) < ∞. Also, inf 1≤i1 <i2 <···<ik ≤mn (1 − pn,i1 ) · · · (1 − pn,ik ) ≥ (1 − μn )k → 1 as n → ∞. (1.20) Finally, with i1 ,...,ik ∈{1,2,...,mn } denoting summation over all ordered ktuples of distinct elements of {1, 2, . . . , mn }, we have pn,i1 pn,i2 · · · pn,ik = pn,i1 pn,i2 · · · pn,ik , k! 1≤i1 <i2 <···<ik ≤mn and i1 ,...,ik ∈{1,2,...,mn } ⎛ mn ⎞ ⎜⎜⎜ ⎟⎟⎟k 0 ≤ ⎜⎜⎝ pn,i ⎟⎟⎠ − pn,i1 pn,i2 · · · pn,ik i1 ,...,ik ∈{1,2,...,mn } i=1 ⎛m ⎞k−2 m n ⎟⎟⎟ k n 2 ⎜⎜⎜⎜ ≤ pn,i ⎜⎜⎝ pn, j ⎟⎟⎟⎠ , 2 i=1 j=1 which tends to zero as n → ∞. Therefore k! pn,i1 pn,i2 · · · pn,ik → γk as n → ∞. (1.21) 1≤i1 <i2 <···<ik ≤mn The result follows from (1.18) by using (1.19), (1.20) and (1.21). 1.4 The Negative Binomial Distribution A random element Z of N0 is said to have a negative binomial distribution with parameters r > 0 and p ∈ (0, 1] if P(Z = n) = Γ(n + r) (1 − p)n pr , Γ(n + 1)Γ(r) n ∈ N0 , (1.22) 6 Poisson and Other Discrete Distributions where the Gamma function Γ : (0, ∞) → (0, ∞) is deﬁned by ∞ ta−1 e−t dt, a > 0. Γ(a) := (1.23) 0 (In particular Γ(a) = (a−1)! for a ∈ N.) This can be seen to be a probability distribution by Taylor expansion of (1 − x)−r evaluated at x = 1 − p. The probability generating function of Z is given by E sZ ] = pr (1 − s + sp)−r , s ∈ [0, 1]. (1.24) For r ∈ N, such a Z may be interpreted as the number of failures before the rth success in a sequence of independent Bernoulli trials. In the special case r = 1 we get the geometric distribution P(Z = n) = (1 − p)n p, n ∈ N0 . (1.25) Another interesting special case is r = 1/2. In this case P(Z = n) = (2n − 1)!! (1 − p)n p1/2 , 2n n! n ∈ N0 , (1.26) where we recall the deﬁnition (B.6) for (2n − 1)!!. This follows from the √ fact that Γ(n + 1/2) = (2n − 1)!! 2−n π, n ∈ N0 . The negative binomial distribution arises as a mixture of Poisson distributions. To explain this, we need to introduce the Gamma distribution with shape parameter a > 0 and scale parameter b > 0. This is a probability measure on R+ with Lebesgue density x → ba Γ(a)−1 xa−1 e−bx (1.27) on R+ . If a random variable Y has this distribution, then one says that Y is Gamma distributed with shape parameter a and scale parameter b. In this case Y has Laplace transform b a E e−tY = , t ≥ 0. (1.28) b+t In the case a = 1 we obtain the exponential distribution with parameter b. Exercise 1.11 asks the reader to prove the following result. Proposition 1.5 Suppose that the random variable Y ≥ 0 is Gamma distributed with shape parameter a > 0 and scale parameter b > 0. Let Z be an N0 valued random variable such that the conditional distribution of Z given Y is Po(Y). Then Z has a negative binomial distribution with parameters a and b/(b + 1). 1.5 Exercises 7 1.5 Exercises Exercise 1.1 Prove equation (1.10). Exercise 1.2 Let X be a random variable taking values in N0 . Assume that there is a γ ≥ 0 such that E[(X)k ] = γk for all k ∈ N0 . Show that X has a Poisson distribution. (Hint: Derive the Taylor series for g(s) := E[sX ] at s0 = 1.) Exercise 1.3 Conﬁrm Proposition 1.3 by showing that E sX tZ−X = e pγ(s−1) e(1−p)γ(t−1) , s, t ∈ [0, 1], using a direct computation and Proposition B.4. Exercise 1.4 (Generalisation of Proposition 1.2) Let m ∈ N and suppose that X1 , . . . , Xm are independent random variables with Poisson distributions Po(γ1 ), . . . , Po(γm ), respectively. Show that X := X1 + · · · + Xm is Poisson distributed with parameter γ := γ1 + · · · + γm . Assuming γ > 0, show moreover for any k ∈ N that P(X1 = k1 , . . . , Xm = km  X = k) = γ k1 γ km k! 1 m ··· k1 ! · · · km ! γ γ (1.29) for k1 + · · · + km = k. This is a multinomial distribution with parameters k and γ1 /γ, . . . , γm /γ. Exercise 1.5 (Generalisation of Proposition 1.3) Let m ∈ N and suppose that Zn , n ∈ N, is a sequence of independent random vectors in Rm with common distribution P(Z1 = ei ) = pi , i ∈ {1, . . . , m}, where ei is the ith unit vector in Rm and p1 + · · · + pm = 1. Let Z have a Poisson distribution with parameter γ, independent of (Z1 , Z2 , . . .). Show that the components of the random vector X := Zj=1 Z j are independent and Poisson distributed with parameters p1 γ, . . . , pm γ. Exercise 1.6 (Bivariate extension of Proposition 1.4) Let γ > 0, δ ≥ 0. Suppose for n ∈ N that mn ∈ N and for 1 ≤ i ≤ mn that pn,i , qn,i ∈ [0, 1) with mi=1n pn,i → γ and mi=1n qn,i → δ, and max1≤i≤mn max{pn,i , qn,i } → 0 as n → ∞. Suppose for n ∈ N that (Xn , Yn ) = mi=1n (Xn,i , Yn,i ), where each (Xn,i , Yn,i ) is a random 2vector whose components are Bernoulli distributed with parameters pn,i , qn,i , respectively, and satisfy Xn,i Yn,i = 0 almost surely. Assume the random vectors (Xn,i , Yn,i ), 1 ≤ i ≤ mn , are independent. Prove that Xn , Yn are asymptotically (as n → ∞) distributed as a pair of indepen 8 Poisson and Other Discrete Distributions dent Poisson variables with parameters γ, δ, i.e. for k, ∈ N0 , γk δ . n→∞ k! ! Exercise 1.7 (Probability of a Poisson variable being even) Suppose X is Poisson distributed with parameter γ > 0. Using the fact that the probability generating function (1.8) extends to s = −1, verify the identity P(X/2 ∈ Z) = (1 + e−2γ )/2. For k ∈ N with k ≥ 3, using the fact that the probability generating function (1.8) extends to a kth complex root of unity, ﬁnd a closedform formula for P(X/k ∈ Z). lim P(Xn = k, Yn = ) = e−(γ+δ) Exercise 1.8 Let γ > 0, and suppose X is Poisson distributed with parameter γ. Suppose f : N → R+ is such that E[ f (X)1+ε ] < ∞ for some ε > 0. Show that E[ f (X + k)] < ∞ for any k ∈ N. Exercise 1.9 Let 0 < γ < γ . Give an example of a random vector (X, Y) with X Poisson distributed with parameter γ and Y Poisson distributed with parameter γ , such that Y−X is not Poisson distributed. (Hint: First consider a pair X , Y such that Y −X is Poisson distributed, and then modify ﬁnitely many of the values of their joint probability mass function.) Exercise 1.10 Suppose n ∈ N and set [n] := {1, . . . , n}. Suppose that Z is a uniform random permutation of [n], that is a random element of the space Σn of all bijective mappings from [n] to [n] such that P(Z = π) = 1/n! for each π ∈ Σn . For a ∈ R let a := min{k ∈ Z : k ≥ a}. Let γ ∈ [0, 1] and let Xn := card{i ∈ [γn] : Z(i) = i} be the number of ﬁxed points of Z among the ﬁrst γn integers. Show that the distribution of Xn converges to Po(γ), that is γk lim P(Xn = k) = e−γ , k ∈ N0 . n→∞ k! (Hint: Establish an explicit formula for P(Xn = k), starting with the case k = 0.) Exercise 1.11 Prove Proposition 1.5. Exercise 1.12 Let γ > 0 and δ > 0. Find a random vector (X, Y) such that X, Y and X + Y are Poisson distributed with parameter γ, δ and γ + δ, respectively, but X and Y are not independent. 2 Point Processes A point process is a random collection of at most countably many points, possibly with multiplicities. This chapter deﬁnes this concept for an arbitrary measurable space and provides several criteria for equality in distribution. 2.1 Fundamentals The idea of a point process is that of a random, at most countable, collection Z of points in some space X. A good example to think of is the ddimensional Euclidean space Rd . Ignoring measurability issues for the moment, we might think of Z as a mapping ω → Z(ω) from Ω into the system of countable subsets of X, where (Ω, F , P) is an underlying probability space. Then Z can be identiﬁed with the family of mappings ω → η(ω, B) := card(Z(ω) ∩ B), B ⊂ X, counting the number of points that Z has in B. (We write card A for the number of elements of a set A.) Clearly, for any ﬁxed ω ∈ Ω the mapping η(ω, ·) is a measure, namely the counting measure supported by Z(ω). It turns out to be a mathematically fruitful idea to deﬁne point processes as random counting measures. To give the general deﬁnition of a point process let (X, X) be a measurable space. Let N<∞ (X) ≡ N<∞ denote the space of all measures μ on X such that μ(B) ∈ N0 := N ∪ {0} for all B ∈ X, and let N(X) ≡ N be the space of all measures that can be written as a countable sum of measures from N<∞ . A trivial example of an element of N is the zero measure 0 that is identically zero on X. A less trivial example is the Dirac measure δ x at a point x ∈ X given by δ x (B) := 1B (x). More generally, any (ﬁnite or inﬁnite) sequence (xn )kn=1 of elements of X, where k ∈ N := N ∪ {∞} is the number 9 Point Processes 10 of terms in the sequence, can be used to deﬁne a measure μ= k δ xn . (2.1) n=1 Then μ ∈ N and μ(B) = k 1B (xn ), B ∈ X. n=1 More generally we have, for any measurable f : X → [0, ∞], that f dμ = k f (xn ). (2.2) n=1 We can allow for k = 0 in (2.1). In this case μ is the zero measure. The points x1 , x2 , . . . are not assumed to be pairwise distinct. If xi = x j for some i, j ≤ k with i j, then μ is said to have multiplicities. In fact, the multiplicity of xi is the number card{ j ≤ k : x j = xi }. Any μ of the form (2.1) is interpreted as a counting measure with possible multiplicities. In general one cannot guarantee that any μ ∈ N can be written in the form (2.1); see Exercise 2.5. Fortunately, only weak assumptions on (X, X) and μ are required to achieve this; see e.g. Corollary 6.5. Moreover, large parts of the theory can be developed without imposing further assumptions on (X, X), other than to be a measurable space. A measure ν on X is said to be sﬁnite if ν is a countable sum of ﬁnite measures. By deﬁnition, each element of N is sﬁnite. We recall that a measure ν on X is said to be σﬁnite if there is a sequence Bm ∈ X, m ∈ N, such that ∪m Bm = X and ν(Bm ) < ∞ for all m ∈ N. Clearly every σﬁnite measure is sﬁnite. Any N0 valued σﬁnite measure is in N. In contrast to σﬁnite measures, any countable sum of sﬁnite measures is again sﬁnite. If the points xn in (2.1) are all the same, then this measure μ is not σﬁnite. The counting measure on R (supported by R) is an example of a measure with values in N0 := N ∪ {0}, that is not sﬁnite. Exercise 6.10 gives an example of an sﬁnite N0 valued measure that is not in N. Let N(X) ≡ N denote the σﬁeld generated by the collection of all subsets of N of the form {μ ∈ N : μ(B) = k}, B ∈ X, k ∈ N0 . This means that N is the smallest σﬁeld on N such that μ → μ(B) is measurable for all B ∈ X. 2.1 Fundamentals 11 Deﬁnition 2.1 A point process on X is a random element η of (N, N), that is a measurable mapping η : Ω → N. If η is a point process on X and B ∈ X, then we denote by η(B) the mapping ω → η(ω, B) := η(ω)(B). By the deﬁnitions of η and the σﬁeld N these are random variables taking values in N0 , that is {η(B) = k} ≡ {ω ∈ Ω : η(ω, B) = k} ∈ F , B ∈ X, k ∈ N0 . (2.3) Conversely, a mapping η : Ω → N is a point process if (2.3) holds. In this case we call η(B) the number of points of η in B. Note that the mapping (ω, B) → η(ω, B) is a kernel from Ω to X (see Section A.1) with the additional property that η(ω, ·) ∈ N for each ω ∈ Ω. Example 2.2 Let X be a random element in X. Then η := δX is a point process. Indeed, the required measurability property follows from ⎧ ⎪ ⎪ {X ∈ B}, if k = 1, ⎪ ⎪ ⎪ ⎨ {η(B) = k} = ⎪ {X B}, if k = 0, ⎪ ⎪ ⎪ ⎪ ⎩∅, otherwise. The above onepoint process can be generalised as follows. Example 2.3 Let Q be a probability measure on X and suppose that X1 , . . . , Xm are independent random elements in X with distribution Q. Then η := δX1 + · · · + δXm is a point process on X. Because m P(η(B) = k) = Q(B)k (1 − Q(B))m−k , k k = 0, . . . , m, η is referred to as a binomial process with sample size m and sampling distribution Q. In this example, the random measure η can be written as a sum of Dirac measures, and we formalise the class of point processes having this property in the following deﬁnition. Here and later we say that two point processes η and η are almost surely equal if there is an A ∈ F with P(A) = 1 such that η(ω) = η (ω) for each ω ∈ A. Point Processes 12 Deﬁnition 2.4 We shall refer to a point process η on X as a proper point process if there exist random elements X1 , X2 , . . . in X and an N0 valued random variable κ such that almost surely η= κ δ Xn . (2.4) n=1 In the case κ = 0 this is interpreted as the zero measure on X. The motivation for this terminology is that the intuitive notion of a point process is that of a (random) set of points, rather than an integervalued measure. A proper point process is one which can be interpreted as a countable (random) set of points in X (possibly with repetitions), thereby better ﬁtting this intuition. The class of proper point processes is very large. Indeed, we shall see later that if X is a Borel subspace of a complete separable metric space, then any locally ﬁnite point process on X (see Deﬁnition 2.13) is proper, and that, for general (X, X), if η is a Poisson point process on X there is a proper point process on X having the same distribution as η (these concepts will be deﬁned in due course); see Corollary 6.5 and Corollary 3.7. Exercise 2.5 shows, however, that not all point processes are proper. 2.2 Campbell’s Formula A ﬁrst characteristic of a point process is the mean number of points lying in an arbitrary measurable set: Deﬁnition 2.5 The intensity measure of a point process η on X is the measure λ deﬁned by λ(B) := E[η(B)], B ∈ X. (2.5) It follows from basic properties of expectation that the intensity measure of a point process is indeed a measure. Example 2.6 The intensity measure of a binomial process with sample size m and sampling distribution Q is given by m m λ(B) = E 1{Xk ∈ B} = P(Xk ∈ B) = m Q(B). k=1 k=1 Independence of the random variables X1 , . . . , Xm is not required for this calculation. 2.2 Campbell’s Formula 13 Let R := [−∞, ∞] and R+ := [0, ∞]. Let us denote by R(X) (resp. R(X)) the set of all measurable functions u : X → R (resp. u : X → R). Let R+ (X) (resp. R+ (X)) be the set of all those u ∈ R(X) (resp. u ∈ R(X)) with u ≥ 0. Given u ∈ R(X), deﬁne the functions u+ , u− ∈ R+ (X) by u+ (x) := max{u(x), 0} and u− (x) := max{−u(x), 0}, x ∈ X. Then u(x) = u+ (x)−u− (x). We recall from measure theory (see Section A.1) that, for any measure ν on X, the integral u dν ≡ u(x) ν(dx) of u ∈ R(X) with respect to ν is deﬁned as + u(x) ν(dx) ≡ u dν := u dν − u− dν whenever this expression is not of the form ∞ − ∞. Otherwise we use here the convention u(x) ν(dx) := 0. We often write ν(u) := u(x) ν(dx), so that ν(B) = ν(1B ) for any B ∈ X. If η is a point process, then η(u) ≡ u dη denotes the mapping ω → u(x) η(ω, dx). Proposition 2.7 (Campbell’s formula) Let η be a point process on (X, X) with intensity measure λ. Let u ∈ R(X). Then u(x) η(dx) is a random variable. Moreover, E u(x) η(dx) = u(x) λ(dx) (2.6) whenever u ≥ 0 or u(x) λ(dx) < ∞. Proof If u(x) = 1B (x) for some B ∈ X then u(x) η(dx) = η(B) and both assertions are true by deﬁnition. By standard techniques of measure theory (linearity and monotone convergence) this can be extended, ﬁrst to measurable simple functions and then to arbitrary u ∈ R+ (X). Let u ∈ R(X). We have just seen that η(u+ ) and η(u−) are random variables, so that η(u) is a random variable too. Assume that u(x) λ(dx) < ∞. Then the ﬁrst part of the proof shows that η(u+ ) and η(u− ) both have a ﬁnite expectation and that E[η(u)] = E[η(u+ )] − E[η(u− )] = λ(u+ ) − λ(u− ) = λ(u). This concludes the proof. Point Processes 14 2.3 Distribution of a Point Process In accordance with the terminology of probability theory (see Section B.1), the distribution of a point process η on X is the probability measure Pη on (N, N), given by A → P(η ∈ A). If η is another point process with the d same distribution, we write η = η . The following device is a powerful tool for analysing point processes. We use the convention e−∞ := 0. Deﬁnition 2.8 The Laplace (or characteristic) functional of a point process η on X is the mapping Lη : R+ (X) → [0, 1] deﬁned by Lη (u) := E exp − u(x) η(dx) , u ∈ R+ (X). Example 2.9 u ∈ R+ (X), Let η be the binomial process of Example 2.3. Then, for m Lη (u) = E exp − u(Xk ) = E k=1 m = E exp[−u(Xk )] = exp[−u(Xk )] m k=1 m exp[−u(x)] Q(dx) . k=1 The following proposition characterises equality in distribution for point processes. It shows, in particular, that the Laplace functional of a point process determines its distribution. Proposition 2.10 For point processes η and η on X the following assertions are equivalent: d (i) η = η ; d (ii) (η(B1 ), . . . , η(Bm )) = (η (B1 ), . . . , η (Bm )) for all m ∈ N and all pairwise disjoint B1 , . . . , Bm ∈ X; (iii) Lη (u) = Lη (u) for all u ∈ R+ (X); d (iv) for all u ∈ R+ (X), η(u) = η (u) as random variables in R+ . Proof First we prove that (i) implies (iv). Given u ∈ R+ (X), deﬁne the function gu : N → R+ by μ → u dμ. By Proposition 2.7 (or a direct check based on ﬁrst principles), gu is a measurable function. Also, Pη(u) (·) = P(η(u) ∈ ·) = P(η ∈ g−1 u (·)), d d and likewise for η . So if η = η then also η(u) = η (u). Next we show that (iv) implies (iii). For any R+ valued random variable 2.3 Distribution of a Point Process 15 Y we have E[exp(−Y)] = e−y PY (dy), which is determined by the distribution PY . Hence, if (iv) holds, Lη (u) = E[exp(−η(u))] = E[exp(−η (u))] = Lη (u) for all u ∈ R+ (X), so (iii) holds. Assume now that (iii) holds and consider a simple function of the form u = c1 1B1 + · · · + cm 1Bm , where m ∈ N, B1 , . . . , Bm ∈ X and c1 , . . . , cm ∈ (0, ∞). Then m Lη (u) = E exp − c j η(B j ) = P̂(η(B1 ),...,η(Bm )) (c1 , . . . , cm ) (2.7) j=1 where for any measure μ on [0, ∞]m we write μ̂ for its multivariate Laplace transform. Since a ﬁnite measure on Rm+ is determined by its Laplace transform (this follows from Proposition B.4), we can conclude that the restriction of P(η(B1 ),...,η(Bm )) (a measure on [0, ∞]m ) to (0, ∞)m is the same as the restriction of P(η (B1 ),...,η (Bm )) to (0, ∞)m . Then, using the fact that P(η(B1 ),...,η(Bm )) and P(η (B1 ),...,η (Bm )) are probability measures on [0, ∞]m , by forming suitable complements we obtain P(η(B1 ),...,η(Bm )) = P(η (B1 ),...,η (Bm )) (these details are left to the reader). In other words, (iii) implies (ii). Finally we assume (ii) and prove (i). Let m ∈ N and B1 , . . . , Bm ∈ X, not necessarily pairwise disjoint. Let C1 , . . . , Cn be the atoms of the ﬁeld generated by B1 , . . . , Bm ; see Section A.1. For each i ∈ {1, . . . , m} there exists Ji ⊂ {1, . . . , n} such that Bi = ∪ j∈Ji C j . (Note that Ji = ∅ if Bi = ∅.) Let D1 , . . . , Dm ⊂ N0 . Then P(η(B1 ) ∈ D1 , . . . , η(Bm ) ∈ Dm ) = 1 k j ∈ D1 , . . . , k j ∈ Dm P(η(C1 ),...,η(Cn )) )(d(k1 , . . . , kn )). j∈J1 j∈Jm Therefore Pη and Pη coincide on the system H consisting of all sets of the form {μ ∈ N : μ(B1 ) ∈ D1 , . . . , μ(Bm ) ∈ Dm }, where m ∈ N, B1 , . . . , Bm ∈ X and D1 , . . . , Dm ⊂ N0 . Clearly H is a πsystem; that is, closed under pairwise intersections. Moreover, the smallest σﬁeld σ(H) containing H is the full σﬁeld N. Hence (i) follows from the fact that a probability measure is determined by its values on a generating πsystem; see Theorem A.5. Point Processes 16 2.4 Point Processes on Metric Spaces Let us now assume that X is a metric space with metric ρ; see Section A.2. Then it is always to be understood that X is the Borel σﬁeld B(X) of X. In particular, the singleton {x} is in X for all x ∈ X. If ν is a measure on X then we often write ν{x} := ν({x}). If ν{x} = 0 for all x ∈ X, then ν is said to be diﬀuse. Moreover, if μ ∈ N(X) then we write x ∈ μ if μ({x}) > 0. A set B ⊂ X is said to be bounded if it is empty or its diameter d(B) := sup{ρ(x, y) : x, y ∈ B} is ﬁnite. Deﬁnition 2.11 Suppose that X is a metric space. The system of bounded measurable subsets of X is denoted by Xb . A measure ν on X is said to be locally ﬁnite if ν(B) < ∞ for every B ∈ Xb . Let Nl (X) denote the set of all locally ﬁnite elements of N(X) and let Nl (X) := {A ∩ Nl (X) : A ∈ N(X)}. Fix some x0 ∈ X. Then any bounded set B is contained in the closed ball B(x0 , r) = {x ∈ X : ρ(x, x0 ) ≤ r} for suﬃciently large r > 0. In fact, if B ∅, then we can take, for instance, r := d(B) + ρ(x1 , x0 ) for some x1 ∈ B. Note that B(x0 , n) ↑ X as n → ∞. Hence a measure ν on X is locally ﬁnite if and only if ν(B(x0 , n)) < ∞ for each n ∈ N. In particular, the set Nl (X) is measurable, that is Nl (X) ∈ N(X). Moreover, any locally ﬁnite measure is σﬁnite. Proposition 2.12 Let η and η be point processes on a metric space X. d Suppose η(u) = η (u) for all u ∈ R+ (X) such that {u > 0} is bounded. Then d η = η . Proof Suppose that d η(u) = η (u), u ∈ R+ (X), {u > 0} bounded. (2.8) Then Lη (u) = Lη (u) for any u ∈ R+ (X) such that {u > 0} is bounded. Given any ∈ R+ (X), we can choose a sequence un , n ∈ N, of functions in R+ (X) such that {un > 0} is bounded for each n, and un ↑ pointwise. Then, by dominated convergence and (2.8), Lη () = lim Lη (un ) = lim Lη (un ) = Lη (), n→∞ d so η = η by Proposition 2.10. n→∞ Deﬁnition 2.13 A point process η on a metric space X is said to be locally ﬁnite if P(η(B) < ∞) = 1 for every bounded B ∈ X. 2.4 Point Processes on Metric Spaces 17 If required, we could interpret a locally ﬁnite point process η as a random element of the space (Nl (X), Nl (X)), introduced in Deﬁnition 2.11. Indeed, we can deﬁne another point process η̃ by η̃(ω, ·) := η(ω, ·) if the latter is locally ﬁnite and by η̃(ω, ·) := 0 (the zero measure) otherwise. Then η̃ is a random element of (Nl (X), Nl (X)) that coincides Palmost surely (Pa.s.) with η. The reader might have noticed that the proof of Proposition 2.12 has not really used the metric on X. The proof of the next reﬁnement of this result (not used later in the book) exploits the metric in an essential way. Proposition 2.14 Let η and η be locally ﬁnite point processes on a metric d space X. Suppose η(u) = η (u) for all continuous u : X → R+ such that d {u > 0} is bounded. Then η = η . Proof Let G be the space of continuous functions u : X → R+ such that d {u > 0} is bounded. Assume that η(u) = η (u) for all u ∈ G. Since G is closed under nonnegative linear combinations, it follows, as in the proof that (iii) implies (ii) in Proposition 2.10, that d (η(u1 ), η(u2 ), . . . ) = (η (u1 ), η (u2 ), . . . ), ﬁrst for any ﬁnite sequence and then (by Theorem A.5 in Section A.1) for any inﬁnite sequence un ∈ G, n ∈ N. Take a bounded closed set C ⊂ X and, for n ∈ N, deﬁne un (x) := max{1 − nd(x, C), 0}, x ∈ X, where d(x, C) := inf{ρ(x, y) : y ∈ C} and inf ∅ := ∞. By Exercise 2.8, un ∈ G. Moreover, un ↓ 1C as n → ∞, and since η is locally ﬁnite we obtain η(un ) → η(C) Pa.s. The same relation holds for η . It follows that statement (ii) of Proposition 2.10 holds whenever B1 , . . . , Bm are closed and bounded, but not necessarily disjoint. Hence, ﬁxing a closed ball C ⊂ X, Pη and Pη coincide on the πsystem HC consisting of all sets of the form {μ ∈ Nl : μ(B1 ∩ C) ≤ k1 , . . . , μ(Bm ∩ C) ≤ km }, (2.9) where m ∈ N, B1 , . . . , Bm ⊂ X are closed and k1 , . . . , km ∈ N0 . Another application of Theorem A.5 shows that Pη and Pη coincide on σ(HC ) and then also on Nl := σ(∪∞ i=1 σ(H Bi )), where Bi := B(x0 , i) and x0 ∈ X is ﬁxed. It remains to show that Nl = Nl . Let i ∈ N and let Ni denote the smallest σﬁeld on Nl containing the sets {μ ∈ Nl : μ(B ∩ Bi ) ≤ k} for all closed sets B ⊂ X and each k ∈ N0 . Let D be the system of all Borel sets B ⊂ X such that μ → μ(B∩Bi ) is Ni measurable. Then D is a Dynkin system containing 18 Point Processes the πsystem of all closed sets, so that the monotone class theorem shows D = X. Therefore σ(HBi ) contains {μ ∈ Nl : μ(B ∩ Bi ) ≤ k} for all B ∈ X and all k ∈ N0 . Letting i → ∞ we see that Nl contains {μ ∈ Nl : μ(B) ≤ k} and therefore every set from Nl . 2.5 Exercises Exercise 2.1 Give an example of a point process η on a measurable space (X, X) with intensity measure λ and u ∈ R(X) (violating the condition that u ≥ 0 or u(x)λ(dx) < ∞), such that Campbell’s formula (2.6) fails. Exercise 2.2 Let X∗ ⊂ X be a πsystem generating X. Let η be a point process on X that is σﬁnite on X∗ , meaning that there is a sequence Cn ∈ X∗ , n ∈ N, such that ∪∞ n=1 C n = X and P(η(C n ) < ∞) = 1 for all n ∈ N. Let η be another point process on X and suppose that the equality in Proposition d 2.10(ii) holds for all B1 , . . . , Bm ∈ X∗ and m ∈ N. Show that η = η . Exercise 2.3 Let η1 , η2 , . . . be a sequence of point processes and deﬁne η := η1 + η2 + · · · , that is η(ω, B) := η1 (ω, B) + η2 (ω, B) + · · · for all ω ∈ Ω and B ∈ X. Show that η is a point process. (Hint: Prove ﬁrst that N(X) is closed under countable summation.) Exercise 2.4 Let η1 , η2 , . . . be a sequence of proper point processes. Show that η := η1 + η2 + · · · is a proper point process. Exercise 2.5 Suppose that X = [0, 1]. Find a σﬁeld X and a measure μ on (X, X) such that μ(X) = 1 and μ(B) ∈ {0, 1} for all B ∈ X, which is not of the form μ = δ x for some x ∈ X. (Hint: Take the system of all ﬁnite subsets of X as a generator of X.) Exercise 2.6 Let η be a point process on X with intensity measure λ and let B ∈ X such that λ(B) < ∞. Show that d λ(B) = − Lη (t1B ) . t=0 dt Exercise 2.7 Let η be a point process on X. Show for each B ∈ X that P(η(B) = 0) = lim Lη (t1B ). t→∞ Exercise 2.8 Let (X, ρ) be a metric space. Let C ⊂ X, C ∅. For x ∈ X let d(x, C) := inf{ρ(x, z) : z ∈ C}. Show that d(·, C) has the Lipschitz property d(x, C) − d(y, C) ≤ ρ(x, y), x, y ∈ X. (Hint: Take z ∈ C and bound ρ(x, z) by the triangle inequality.) 3 Poisson Processes For a Poisson point process the number of points in a given set has a Poisson distribution. Moreover, the numbers of points in disjoint sets are stochastically independent. A Poisson process exists on a general sﬁnite measure space. Its distribution is characterised by a speciﬁc exponential form of the Laplace functional. 3.1 Deﬁnition of the Poisson Process In this chapter we ﬁx an arbitrary measurable space (X, X). We are now ready for the deﬁnition of the main subject of this volume. Recall that for γ ∈ [0, ∞], the Poisson distribution Po(γ) was deﬁned at (1.4). Deﬁnition 3.1 Let λ be an sﬁnite measure on X. A Poisson process with intensity measure λ is a point process η on X with the following two properties: (i) For every B ∈ X the distribution of η(B) is Poisson with parameter λ(B), that is to say P(η(B) = k) = Po(λ(B); k) for all k ∈ N0 . (ii) For every m ∈ N and all pairwise disjoint sets B1 , . . . , Bm ∈ X the random variables η(B1 ), . . . , η(Bm ) are independent. Property (i) of Deﬁnition 3.1 is responsible for the name of the Poisson process. A point process with property (ii) is said to be completely independent. (One also says that η has independent increments or is completely random.) For a (locally ﬁnite) point process without multiplicities and a diﬀuse intensity measure (on a complete separable metric space) we shall see in Chapter 6 that the two deﬁning properties of a Poisson process are equivalent. If η is a Poisson process with intensity measure λ then E[η(B)] = λ(B), so that Deﬁnition 3.1 is consistent with Deﬁnition 2.5. In particular, if λ = 0 is the zero measure, then P(η(X) = 0) = 1. 19 Poisson Processes 20 Let us ﬁrst record that for each sﬁnite λ there is at most one Poisson process with intensity measure λ, up to equality in distribution. Proposition 3.2 Let η and η be two Poisson processes on X with the d same sﬁnite intensity measure. Then η = η . Proof The result follows from Proposition 2.10. 3.2 Existence of Poisson Processes In this section we show by means of an explicit construction that Poisson processes exist. Before we can do this, we need to deal with the superposition of independent Poisson processes. Theorem 3.3 (Superposition theorem) Let ηi , i ∈ N, be a sequence of independent Poisson processes on X with intensity measures λi . Then η := ∞ ηi (3.1) i=1 is a Poisson process with intensity measure λ := λ1 + λ2 + · · · . Proof Exercise 2.3 shows that η is a point process. For n ∈ N and B ∈ X, we have by Exercise 1.4 that ξn (B) := ni=1 ηi (B) has a Poisson distribution with parameter ni=1 λi (B). Also ξn (B) converges monotonically to η(B) so by continuity of probability, and the fact that Po(γ; j) is continuous in γ for j ∈ N0 , for all k ∈ N0 we have P(η(B) ≤ k) = lim P(ξn (B) ≤ k) n→∞ ⎛ n k ⎜⎜ = lim Po ⎜⎜⎜⎝ λi (B); n→∞ j=0 i=1 ⎞ ⎛∞ k ⎟⎟⎟ ⎜⎜ j⎟⎟⎠ = Po ⎜⎜⎜⎝ λi (B); j=0 ⎞ ⎟⎟ j⎟⎟⎟⎠ i=1 so that η(B) has the Po(λ(B)) distribution. Let B1 , . . . , Bm ∈ X be pairwise disjoint. Then (ηi (B j ), 1 ≤ j ≤ m, i ∈ N) is a family of independent random variables, so that by the grouping property of independence the random variables i ηi (B1 ), . . . , i ηi (Bm ) are independent. Thus η is completely independent. Now we construct a Poisson process on (X, X) with arbitrary sﬁnite intensity measure. We start by generalising Example 2.3. Deﬁnition 3.4 Let V and Q be probability measures on N0 and X, respectively. Suppose that X1 , X2 , . . . are independent random elements in X 3.2 Existence of Poisson Processes 21 with distribution Q, and let κ be a random variable with distribution V, independent of (Xn ). Then η := κ δ Xk (3.2) k=1 is called a mixed binomial process with mixing distribution V and sampling distribution Q. The following result provides the key for the construction of Poisson processes. Proposition 3.5 Let Q be a probability measure on X and let γ ≥ 0. Suppose that η is a mixed binomial process with mixing distribution Po(γ) and sampling distribution Q. Then η is a Poisson process with intensity measure γ Q. Proof Let κ and (Xn ) be given as in Deﬁnition 3.4. To prove property (ii) of Deﬁnition 3.1 it is no loss of generality to assume that B1 , . . . , Bm are pairwise disjoint measurable subsets of X satisfying ∪mi=1 Bi = X. (Otherwise we can add the complement of this union.) Let k1 , . . . , km ∈ N0 and set k := k1 + · · · + km . Then P(η(B1 ) = k1 , . . . , η(Bm ) = km ) k k = P(κ = k) P 1{X j ∈ B1 } = k1 , . . . , 1{X j ∈ Bm } = km . j=1 j=1 Since the second probability on the right is multinomial, this gives k! γk −γ e Q(B1 )k1 · · · Q(Bm )km k! k1 ! · · · km ! m (γQ(B j ))k j −γQ(B j ) e = . k j! j=1 P(η(B1 ) = k1 , . . . , η(Bm ) = km ) = Summing over k2 , . . . , km shows that η(B1 ) is Poisson distributed with parameter γ Q(B1 ). A similar statement applies to η(B2 ), . . . , η(Bm ). Therefore η(B1 ), . . . , η(Bm ) are independent. Theorem 3.6 (Existence theorem) Let λ be an sﬁnite measure on X. Then there exists a Poisson process on X with intensity measure λ. Proof The result is trivial if λ(X) = 0. Suppose for now that 0 < λ(X) < ∞. On a suitable probability space, assume that κ, X1 , X2 , . . . are independent random elements, with κ taking 22 Poisson Processes values in N0 and each Xi taking values in X, with κ having the Po(λ(X)) distribution and each Xi having λ(·)/λ(X) as its distribution. Here the probability space can be taken to be a suitable product space; see the proof of Corollary 3.7 below. Let η be the mixed binomial process given by (3.2). Then, by Proposition 3.5, η is a Poisson process with intensity measure λ, as required. Now suppose that λ(X) = ∞. There is a sequence λi , i ∈ N, of measures on (X, X) with strictly positive and ﬁnite total measure, such that λ= ∞ i=1 λi . On a suitable (product) probability space, let ηi , i ∈ N, be a sequence of independent Poisson processes with ηi having intensity measure λi . This is possible by the preceding part of the proof. Set η = ∞ i=1 ηi . By the superposition theorem (Theorem 3.3), η is a Poisson process with intensity measure λ, and the proof is complete. A corollary of the preceding proof is that on arbitrary (X, X) every Poisson point process is proper (see Deﬁnition 2.4), up to equality in distribution. Corollary 3.7 Let λ be an sﬁnite measure on X. Then there is a probability space (Ω, F , P) supporting random elements X1 , X2 , . . . in X and κ in N0 , such that κ δ Xn (3.3) η := n=1 is a Poisson process with intensity measure λ. Proof We consider only the case λ(X) = ∞ (the other case is covered by Proposition 3.5). Take the measures λi , i ∈ N, as in the last part of the proof of Theorem 3.6. Let γi := λi (X) and Qi := γi−1 λi . We shall take (Ω, F , P) to be the product of spaces (Ωi , Fi , Pi ), i ∈ N, where each (Ωi , Fi , Pi ) is again an inﬁnite product of probability spaces (Ωi j , Fi j , Pi j ), j ∈ N0 , with Ωi0 := N0 , Pi0 := Po(γi ) and (Ωi j , Fi j , Pi j ) := (X, X, Qi ) for j ≥ 1. On this space we can deﬁne independent random elements κi , i ∈ N, and Xi j , i, j ∈ N, such that κi has distribution Po(γi ) and Xi j has distribution Qi ; see Theorem B.2. The proof of Theorem 3.6 shows how to deﬁne κ, X1 , X2 , . . . in terms of these random variables in a measurable (algorithmic) way. The details are left to the reader. As a consequence of Corollary 3.7, when checking a statement involving only the distribution of a Poisson process η, it is no restriction of generality to assume that η is proper. Exercise 3.9 shows that there are Poisson processes which are not proper. On the other hand, Corollary 6.5 will show 3.3 Laplace Functional of the Poisson Process 23 that any suitably regular point process on a Borel subset of a complete separable metric space is proper. The next result is a converse to Proposition 3.5. Proposition 3.8 Let η be a Poisson process on X with intensity measure λ satisfying 0 < λ(X) < ∞. Then η has the distribution of a mixed binomial process with mixing distribution Po(λ(X)) and sampling distribution Q := λ(X)−1 λ. The conditional distribution P(η ∈ ·  η(X) = m), m ∈ N, is that of a binomial process with sample size m and sampling distribution Q. Proof Let η be a mixed binomial process that has mixing distribution d Po(λ(X)) and sampling distribution Q. Then η = η by Propositions 3.5 and 3.2. This is our ﬁrst assertion. Also, by deﬁnition, P(η ∈ ·  η (X) = m) has the distribution of a binomial process with sample size m and sampling distribution Q, and by the ﬁrst assertion so does P(η ∈ ·  η(X) = m), yielding the second assertion. 3.3 Laplace Functional of the Poisson Process The following characterisation of Poisson processes is of great value for both theory and applications. Theorem 3.9 Let λ be an sﬁnite measure on X and let η be a point process on X. Then η is a Poisson process with intensity measure λ if and only if Lη (u) = exp − 1 − e−u(x) λ(dx) , u ∈ R+ (X). (3.4) Proof Assume ﬁrst that η is a Poisson process with intensity measure λ. Consider ﬁrst the simple function u := c1 1B1 + · · · + cm 1Bm , where m ∈ N, c1 , . . . , cm ∈ (0, ∞) and B1 , . . . , Bm ∈ X are pairwise disjoint. Then m m E[exp[−η(u)]] = E exp − ci η(Bi ) = E exp[−ci η(Bi )] . i=1 i=1 The complete independence and the formula (1.9) for the Laplace transform of the Poisson distribution (this also holds for Po(∞)) yield m m Lη (u) = E exp[−ci η(Bi )] = exp[−λ(Bi )(1 − e−ci )] i=1 i=1 m m −ci = exp − λ(Bi )(1 − e ) = exp − (1 − e−u ) dλ . i=1 i=1 Bi 24 Poisson Processes Since 1 − e−u(x) = 0 for x B1 ∪ · · · ∪ Bm , this is the righthand side of (3.4). For general u ∈ R+ (X), choose simple functions un with un ↑ u as n → ∞. Then, by monotone convergence (Theorem A.6), η(un ) ↑ η(u) as n → ∞, and by dominated convergence for expectations the lefthand side of E[exp[−η(un )]] = exp − 1 − e−un (x) λ(dx) tends to Lη (u). By monotone convergence again (this time for the integral with respect to λ), the righthand side tends to the righthand side of (3.4). Assume now that (3.4) holds. Let η be a Poisson process with intensity measure λ. (By Theorem 3.6, such an η exists.) By the preceding argument, Lη (u) = Lη (u) for all u ∈ R+ (X). Therefore, by Proposition 2.10, d η = η ; that is, η is a Poisson process with intensity measure λ. 3.4 Exercises Exercise 3.1 Use Exercise 1.12 to deduce that there exist a measure space (X, X, λ) and a point process on X satisfying part (i) but not part (ii) of the deﬁnition of a Poisson process (Deﬁnition 3.1). Exercise 3.2 Show that there exist a measure space (X, X, λ) and a point process η on X satisfying part (i) of Deﬁnition 3.1 and part (ii) of that deﬁnition with ‘independent’ replaced by ‘pairwise independent’, such that η is not a Poisson point process. In other words, show that we can have η(B) Poisson distributed for all B ∈ X, and η(A) independent of η(B) for all disjoint pairs A, B ∈ X, but η(A1 ), . . . , η(Ak ) not mutually independent for all disjoint A1 , . . . , Ak ∈ X. Exercise 3.3 Let η be a Poisson process on X with intensity measure λ and let B ∈ X with 0 < λ(B) < ∞. Suppose B1 , . . . , Bn are sets in X forming a partition of B. Show for all k1 , . . . , kn ∈ N0 and m := i ki that n λ(B ) ki m! i P ∩ni=1 {η(Bi ) = ki }  η(B) = m = . k1 !k2 ! · · · kn ! i=1 λ(B) Exercise 3.4 Let η be a Poisson process on X with sﬁnite intensity measure λ and let u ∈ R+ (X). Use the proof of Theorem 3.9 to show that E exp u(x) η(dx) = exp eu(x) − 1 λ(dx) . 3.4 Exercises 25 Exercise 3.5 Let V be a probability measure on N0 with generating funcn tion GV (s) := ∞ n=0 V({n})s , s ∈ [0, 1]. Let η be a mixed binomial process with mixing distribution V and sampling distribution Q. Show that Lη (u) = GV e−u dQ , u ∈ R+ (X). Assume now that V is a Poisson distribution; show that the preceding formula is consistent with Theorem 3.9. Exercise 3.6 Let η be a point process on X. Using the convention e−∞ := 0, the Laplace functional Lη (u) can be deﬁned for any u ∈ R+ (X). Assume now that η is a Poisson process with intensity measure λ. Use Theorem 3.9 to show that κ E u(Xn ) = exp − (1 − u(x)) λ(dx) , (3.5) n=1 for any measurable u : X → [0, 1], where η is assumed to be given by (3.3). The lefthand side of (3.5) is called the probability generating functional of η. It can be deﬁned for any point process (proper or not) by taking the expectation of exp ln u(x) η(dx) . Exercise 3.7 Let η be a Poisson process with ﬁnite intensity measure λ. Show for all f ∈ R+ (N) that ∞ 1 −λ(X) −λ(X) E[ f (η)] = e f (δ x1 + · · · + δ xn ) λn (d(x1 , . . . , xn )). f (0) + e n! n=1 Exercise 3.8 Let η be a Poisson process with sﬁnite intensity measure λ and let f ∈ R+ (N) be such that E[ f (η)] < ∞. Suppose that η is a Poisson process with intensity measure λ such that λ = λ + ν for some ﬁnite measure ν. Show that E[ f (η )] < ∞. (Hint: Use the superposition theorem.) Exercise 3.9 In the setting of Exercise 2.5, show that there is a probability measure λ on (X, X) and a Poisson process η with intensity measure λ such that η is not proper. (Hint: Use Exercise 2.5.) Exercise 3.10 Let 0 < γ < γ . Give an example of two Poisson processes η, η on (0, 1) with intensity measures γλ1 and γ λ1 , respectively (λ1 denoting Lebesgue measure), such that η ≤ η but η − η is not a Poisson process. (Hint: Use Exercise 1.9.) Exercise 3.11 Let η be a Poisson process with intensity measure λ and let B1 , B2 ∈ X satisfy λ(B1 ) < ∞ and λ(B2 ) < ∞. Show that the covariance between η(B1 ) and η(B2 ) is given by Cov[η(B1 ), η(B2 )] = λ(B1 ∩ B2 ). 4 The Mecke Equation and Factorial Measures The Mecke equation provides a way to compute the expectation of integrals, i.e. sums, with respect to a Poisson process, where the integrand can depend on both the point process and the point in the state space. This functional equation characterises a Poisson process. The Mecke identity can be extended to integration with respect to factorial measures, i.e. to multiple sums. Factorial measures can also be used to deﬁne the Janossy measures, thus providing a local description of a general point process. The factorial moment measures of a point process are deﬁned as the expected factorial measures. They describe the probability of the occurrence of points in a ﬁnite number of inﬁnitesimally small sets. 4.1 The Mecke Equation In this chapter we take (X, X) to be an arbitrary measurable space and use the abbreviation (N, N) := (N(X), N(X)). Let η be a Poisson process on X with sﬁnite intensity measure λ and let f ∈ R+ (X × N). The complete independence of η implies for each x ∈ X that, heuristically speaking, η(dx) and the restriction η{x}c of η to X \ {x} are independent. Therefore E[ f (x, η{x}c )] λ(dx), (4.1) E f (x, η{x}c ) η(dx) = where we ignore measurability issues. If P(η({x}) = 0) = 1 for each x ∈ X (which is the case if λ is adiﬀuse measure on a Borel space), then the righthand side of (4.1) equals E[ f (x, η)] λ(dx). (Exercise 6.11 shows a way to extend this to an arbitrary intensity measure.) We show that a proper version of the resulting integral identity holds in general and characterises the Poisson process. This equation is a fundamental tool for analysing the Poisson process and can be used in many speciﬁc calculations. In the special case where X has just a single element, Theorem 4.1 essentially reduces to an earlier result about the Poisson distribution, namely Proposition 1.1. 26 4.1 The Mecke Equation 27 Theorem 4.1 (Mecke equation) Let λ be an sﬁnite measure on X and η a point process on X. Then η is a Poisson process with intensity measure λ if and only if E f (x, η) η(dx) = E[ f (x, η + δ x )] λ(dx) (4.2) for all f ∈ R+ (X × N). Proof Let us start by noting that the mapping (x, μ) → μ + δ x (adding a point x to the counting measure μ) from X × N to N is measurable. Indeed, the mapping (x, μ) → μ(B) + 1B (x) is measurable for all B ∈ X. If η is a Poisson process, then (4.2) is a special case of (4.11) to be proved in Section 4.2. Assume now that (4.2) holds for all measurable f ≥ 0. Let B1 , . . . , Bm be disjoint sets in X with λ(Bi ) < ∞ for each i. For k1 , . . . , km ∈ N0 with k1 ≥ 1 we deﬁne m f (x, μ) = 1B1 (x) 1{μ(Bi ) = ki }, (x, μ) ∈ X × N. i=1 Then E f (x, η) η(dx) = E η(B1 ) m 1{η(Bi ) = ki } = k1 P ∩mi=1 {η(Bi ) = ki } , i=1 with the (measure theory) convention ∞ · 0 := 0. On the other hand, we have for each x ∈ X that E[ f (x, η + δ x )] = 1B1 (x) P(η(B1 ) = k1 − 1, η(B2 ) = k2 , . . . , η(Bm ) = km ) (with ∞ − 1 := ∞) so that, by (4.2), k1 P ∩mi=1 {η(Bi ) = ki } = λ(B1 ) P {η(B1 ) = k1 − 1} ∩ ∩mi=2 {η(Bi ) = ki } . Assume that P ∩mi=2 {η(Bi ) = ki } > 0 and note that otherwise η(B1 ) and the event ∩mi=2 {η(Bi ) = ki } are independent. Putting πk = P η(B1 ) = k  ∩mi=2 {η(Bi ) = ki } , k ∈ N0 , we have kπk = λ(B1 )πk−1 , k ∈ N. Since λ(B1 ) < ∞ this implies π∞ = 0. The only distribution satisfying this recursion is given by πk = Po(λ(B1 ); k), regardless of k2 , . . . , km ; hence η(B1 ) is Po(λ(B1 )) distributed, and independent of ∩mi=2 {η(Bi ) = ki }. Hence, by an induction on m, the variables η(B1 ), . . . , η(Bm ) are independent. 28 The Mecke Equation and Factorial Measures For general B ∈ X we still get for all k ∈ N that k P(η(B) = k) = λ(B) P(η(B) = k − 1). If λ(B) = ∞ we obtain P(η(B) = k − 1) = 0 and hence P(η(B) = ∞) = 1. It follows that η has the deﬁning properties of the Poisson process. 4.2 Factorial Measures and the Multivariate Mecke Equation Equation (4.2) admits a useful generalisation involving multiple integration. To formulate this version we consider, for m ∈ N, the mth power (Xm , Xm ) of (X, X); see Section A.1. Suppose μ ∈ N is given by μ= k δx j (4.3) j=1 for some k ∈ N0 and some x1 , x2 , . . . ∈ X (not necessarily distinct) as in (2.1). Then we deﬁne another measure μ(m) ∈ N(Xm ) by 1{(xi1 , . . . , xim ) ∈ C}, C ∈ Xm , (4.4) μ(m) (C) = i1 ,...,im ≤k where the superscript indicates summation over mtuples with pairwise diﬀerent entries and where an empty sum is deﬁned as zero. (In the case k = ∞ this involves only integervalued indices.) In other words this means that μ(m) = δ(xi1 ,...,xim ) . (4.5) i1 ,...,im ≤k To aid understanding, it is helpful to consider in (4.4) a set C of the special product form B1 × · · · × Bm . If these sets are pairwise disjoint, then the righthand side of (4.4) factorises, yielding m μ (B1 × · · · × Bm ) = μ(B j ). (m) (4.6) j=1 If, on the other hand, B j = B for all j ∈ {1, . . . , m} then, clearly, μ(m) (Bm ) = μ(B)(μ(B) − 1) · · · (μ(B) − m + 1) = (μ(B))m . (4.7) Therefore μ(m) is called the mth factorial measure of μ. For m = 2 and arbitrary B1 , B2 ∈ X we obtain from (4.4) that μ(2) (B1 × B2 ) = μ(B1 )μ(B2 ) − μ(B1 ∩ B2 ), (4.8) 4.2 Factorial Measures and the Multivariate Mecke Equation 29 provided that μ(B1 ∩ B2 ) < ∞. Otherwise μ(2) (B1 × B2 ) = ∞. Factorial measures satisfy the following useful recursion: Let μ ∈ N be given by (4.3) and deﬁne μ(1) := μ. Then, for Lemma 4.2 all m ∈ N, μ(m+1) = − 1{(x1 , . . . , xm+1 ) ∈ ·} μ(dxm+1 ) m 1{(x1 , . . . , xm , x j ) ∈ ·} μ(m) (d(x1 , . . . , xm )). (4.9) j=1 Proof Let m ∈ N and C ∈ Xm+1 . Then μ(m+1) (C) = i1 ,...,im ≤k k 1{(xi1 , . . . , xim , x j ) ∈ C}. j=1 j{i1 ,...,im } Here the inner sum equals k j=1 1{(xi1 , . . . , xim , x j ) ∈ C} − m 1{(xi1 , . . . , xim , xil ) ∈ C}, l=1 where the latter diﬀerence is either a nonnegative integer (if the ﬁrst sum is ﬁnite) or ∞ (if the ﬁrst sum is inﬁnite). This proves the result. For a general space (X, X) there is no guarantee that a measure μ ∈ N can be represented as in (4.3); see Exercise 2.5. Equation (4.9) suggests a recursive deﬁnition of the factorial measures of a general μ ∈ N, without using a representation as a sum of Dirac measures. The next proposition conﬁrms this idea. Proposition 4.3 For any μ ∈ N there is a unique sequence μ(m) ∈ N(Xm ), m ∈ N, satisfying μ(1) := μ and the recursion (4.9). The mappings μ → μ(m) are measurable. The proof of Proposition 4.3 is given in Section A.1 (see Proposition A.18) and can be skipped without too much loss. It is enough to remember that μ(m) can be deﬁned by (4.4), whenever μ is given by (4.3). This follows from Lemma 4.2 and the fact that the solution of (4.9) must be unique. It follows by induction that (4.6) and (4.7) remain valid for general μ ∈ N; see Exercise 4.4. Let η be a point process on X and let m ∈ N. Proposition 4.3 shows that The Mecke Equation and Factorial Measures 30 η(m) is a point process on Xm . If η is proper and given as at (2.4), then δ(Xi1 ,...,Xim ) . (4.10) η(m) = i1 ,...,im ∈{1,...,κ} We continue with the multivariate version of the Mecke equation (4.2). Theorem 4.4 (Multivariate Mecke equation) Let η be a Poisson process on X with sﬁnite intensity measure λ. Then, for every m ∈ N and for every f ∈ R+ (Xm × N), (m) E f (x1 , . . . , xm , η) η (d(x1 , . . . , xm )) = E f (x1 , . . . , xm , η + δ x1 + · · · + δ xm ) λm (d(x1 , . . . , xm )). (4.11) Proof By Proposition 4.3, the map μ → μ(m) is measurable, so that (4.11) involves only the distribution of η. By Corollary 3.7 we can hence assume that η is proper and given by (2.4). Let us ﬁrst assume that λ(X) < ∞. Then λ = γ Q for some γ ≥ 0 and some probability measure Q on X. By Proposition 3.5, we can then assume that η is a mixed binomial process as in Deﬁnition 3.4, with κ having the Po(γ) distribution. Let f ∈ R+ (Xm × N). Then we obtain from (4.10) and (2.2) that the lefthand side of (4.11) equals e−γ ∞ γk E f (Xi1 , . . . , Xim , δX1 + · · · + δXk ) k! i ,...,i ∈{1,...,k} k=m 1 = e−γ ∞ γk k=m k! m E f (Xi1 , . . . , Xim , δX1 + · · · + δXk ) , (4.12) i1 ,...,im ∈{1,...,k} where we have used ﬁrst independence of κ and (Xn ) and then the fact that we can perform integration and summation in any order we want (since f ≥ 0). Let us denote by y = (y1 , . . . , ym ) a generic element of Xm . Since the Xi are independent with distribution Q, the expression (4.12) equals k−m m ∞ γk (k)m e−γ E f y, δ Xi + δy j Qm (dy) k! i=1 j=1 k=m ∞ k−m m γk−m −γ m E f y, δXi + δy j Qm (dy) =e γ (k − m)! i=1 j=1 k=m = E f (y1 , . . . , ym , η + δy1 + · · · + δym ) λm (d(y1 , . . . , ym )), 4.2 Factorial Measures and the Multivariate Mecke Equation 31 where we have again used the mixed binomial representation. This proves (4.11) for ﬁnite λ. Now suppose λ(X) = ∞. As in the proof of Theorem 3.6 we can then assume that η = i ηi , where ηi are independent proper Poisson processes with intensity measures λi each having ﬁnite total measure. By the grouping property of independence, the point processes η j , χi := ηj ξi := j≤i j≥i+1 are independent for each i ∈ N. By (4.10) we have ξi(m) ↑ η(m) as i → ∞. Hence we can apply monotone convergence (Theorem A.12) to see that the lefthand side of (4.11) is given by lim E f (x1 , . . . , xm , ξi + χi ) ξi(m) (d(x1 , . . . , xm )) i→∞ = lim E fi (x1 , . . . , xm , ξi ) ξi(m) (d(x1 , . . . , xm )) , (4.13) i→∞ where fi (x1 , . . . , xm , μ) := E f (x1 , . . . , xm , μ + χi ) , (x1 , . . . , xm , μ) ∈ Xm × N. Setting λi := ij=1 λ j , we can now apply the previous result to obtain from Fubini’s theorem (Theorem A.13) that the expression (4.13) equals lim E[ fi (x1 , . . . , xm , ξi + δ x1 + · · · + δ xm )] (λi )m (d(x1 , . . . , xm )) i→∞ = lim E f (x1 , . . . , xm , η + δ x1 + · · · + δ xm ) (λi )m (d(x1 , . . . , xm )). i→∞ By (A.7) this is the righthand side of (4.11). Next we formulate another useful version of the multivariate Mecke equation. For μ ∈ N and x ∈ X we deﬁne the measure μ \ δ x ∈ N by ⎧ ⎪ ⎪ ⎨μ − δ x , if μ ≥ δ x , μ \ δ x := ⎪ (4.14) ⎪ ⎩μ, otherwise. For x1 , . . . , xm ∈ X, the measure μ \ δ x1 \ · · · \ δ xm ∈ N is deﬁned inductively. Theorem 4.5 Let η be a proper Poisson process on X with sﬁnite intensity measure λ and let m ∈ N. Then, for any f ∈ R+ (Xm × N), E f (x1 , . . . , xm , η \ δ x1 \ · · · \ δ xm ) η(m) (d(x1 , . . . , xm )) = E[ f (x1 , . . . , xm , η)] λm (d(x1 , . . . , xm )). (4.15) 32 The Mecke Equation and Factorial Measures Proof If X is a subspace of a complete separable metric space as in Proposition 6.2, then it is easy to show that (x1 , . . . , xm , μ) → μ \ δ x1 \ · · · \ δ xm is a measurable mapping from Xm × Nl (X) to Nl (X). In that case, and if λ is locally ﬁnite, (4.15) follows upon applying (4.11) to the function (x1 , . . . , xm , μ) → f (x1 , . . . , xm , μ \ δ x1 \ · · · \ δ xm ). In the general case we use that η is proper. Therefore the mapping (ω, x1 , . . . , xm ) → η(ω)\δ x1 \· · ·\δ xm is measurable, which is enough to make (4.15) a meaningful statement. The proof can proceed in exactly the same way as the proof of Theorem 4.4. 4.3 Janossy Measures The restriction νB of a measure ν on X to a set B ∈ X is a measure on X deﬁned by νB (B ) := ν(B ∩ B ), B ∈ X. (4.16) If η is a point process on X, then so is its restriction ηB . For B ∈ X, m ∈ N and a measure ν on X we write νmB := (νB )m . For a point process η on X we (m) write η(m) B := (η B ) . Factorial measures can be used to describe the restriction of point processes as follows. Deﬁnition 4.6 Let η be a point process on X, let B ∈ X and m ∈ N. The Janossy measure of order m of η restricted to B is the measure on Xm deﬁned by Jη,B,m := 1 E 1{η(B) = m}η(m) B (·) . m! (4.17) The number Jη,B,0 := P(η(B) = 0) is called the Janossy measure of order 0. Note that the Janossy measures Jη,B,m are symmetric (see (A.17))) and Jη,B,m (Xm ) = P(η(B) = m), m ∈ N. (4.18) If P(η(B) < ∞) = 1, then the Janossy measures determine the distribution of the restriction ηB of η to B: Theorem 4.7 Let η and η be point processes on X. Let B ∈ X and assume that Jη,B,m = Jη ,B,m for each m ∈ N0 . Then P(η(B) < ∞, ηB ∈ ·) = P(η (B) < ∞, ηB ∈ ·). 4.3 Janossy Measures 33 Proof For notational convenience we assume that B = X. Let m ∈ N and suppose that μ ∈ N satisﬁes μ(X) = m. We assert for each A ∈ N that ! " 1 1 δ x1 + · · · + δ xm ∈ A μ(m) (d(x1 , . . . , xm )). (4.19) 1{μ ∈ A} = m! Since both sides of (4.19) are ﬁnite measures in A, it suﬃces to prove this identity for each set A of the form A = {ν ∈ N : ν(B1 ) = i1 , . . . , ν(Bn ) = in }, where n ∈ N, B1 , . . . , Bn ∈ X and i1 , . . . , in ∈ N0 . Given such a set, let μ be deﬁned as in Lemma A.15. Then μ ∈ A if and only if μ ∈ A and the righthand side of (4.19) does not change upon replacing μ by μ . Hence it suﬃces to check (4.19) for ﬁnite sums of Dirac measures. This is obvious from (4.4). It follows from (4.17) that for all m ∈ N and f ∈ R+ (X) we have 1 f dJη,X,m = E 1{η(B) = m} f dη(m) . (4.20) m! From (4.19) and (4.20) we obtain for each A ∈ N that P(η(X) < ∞, η ∈ A) = 1{0 ∈ A}Jη,X,0 + ∞ 1{δ x1 + · · · + δ xm ∈ A} Jη,X,m (d(x1 , . . . , xm )) m=1 and hence the assertion. Example 4.8 Let η be a Poisson process on X with sﬁnite intensity measure λ. Let m ∈ N and B ∈ X. By the multivariate Mecke equation (Theorem 4.4) we have for each C ∈ Xm that 1 E 1{η(B) = m}η(m) (Bm ∩ C) m! 1 = E 1{(η + δ x1 + · · · + δ xm )(B) = m} λmB (d(x1 , . . . , xm )) . m! C Jη,B,m (C) = For x1 , . . . , xm ∈ B we have (η + δ x1 + · · · + δ xm )(B) = m if and only if η(B) = 0. Therefore we obtain Jη,B,m = e−λ(B) m λ , m! B m ∈ N. (4.21) The Mecke Equation and Factorial Measures 34 4.4 Factorial Moment Measures Deﬁnition 4.9 For m ∈ N the mth factorial moment measure of a point process η is the measure αm on Xm deﬁned by αm (C) := E[η(m) (C)], C ∈ Xm . (4.22) If the point process η is proper, i.e. given by (2.4), then αm (C) = E 1{(Xi1 , . . . , Xim ) ∈ C} , i1 ,...,im ≤κ and hence for f ∈ R+ (Xm ) we have that f (x1 , . . . , xm ) αm (d(x1 , . . . , xm )) = E i1 ,...,im ≤κ Xm (4.23) f (Xi1 , . . . , Xim ) . The ﬁrst factorial moment measure of a point process η is just the intensity measure of Deﬁnition 2.5, while the second describes the second order properties of η. For instance, it follows from (4.8) (and Exercise 4.4 if η is not proper) that α2 (B1 × B2 ) = E[η(B1 )η(B2 )] − E[η(B1 ∩ B2 )], (4.24) provided that E[η(B1 ∩ B2 )] < ∞. Theorem 4.4 has the following immediate consequence: Corollary 4.10 Given m ∈ N the mth factorial moment measure of a Poisson process with sﬁnite intensity measure λ is λm . Proof Apply (4.11) to the function f (x1 , . . . , xm , η) = 1{(x1 , . . . , xm ) ∈ C} for C ∈ Xm . Let η be a point process on X with intensity measure λ and let f, g ∈ L1 (λ) ∩ L2 (λ). By the Cauchy–Schwarz inequality ((A.2) for p = q = 2) we have f g ∈ L1 (λ) so that Campbell’s formula (Proposition 2.7) shows that η( f ) < ∞ and η( f g) < ∞ hold almost surely. Therefore it follows from the case m = 1 of (4.9) that f (x) f (y) η(2) (d(x, y)) = η( f )η(g) − η( f g), Pa.s. Reordering terms and taking expectations gives E[η( f )η(g)] = λ( f g) + f (x)g(y) α2 (d(x, y)), provided that (4.25)  f (x)g(y) α2 (d(x, y)) < ∞ or f, g ≥ 0. If η is a Poisson 4.4 Factorial Moment Measures 35 process with sﬁnite intensity measure λ, then (4.25) and Corollary 4.10 imply the following useful generalisation of Exercise 3.11: f, g ∈ L1 (λ) ∩ L2 (λ). E[η( f )η(g)] = λ( f g) + λ( f )λ(g), (4.26) Under certain assumptions the factorial moment measures of a point process determine its distribution. To derive this result we need the following lemma. We use the conventions e−∞ := 0 and log 0 := −∞. Lemma 4.11 Let η be a point process on X. Let B ∈ X and assume that there exists c > 1 such that the factorial moment measures αn of η satisfy αn (Bn ) ≤ n!cn , n ≥ 1. (4.27) Let u ∈ R+ (X) and a < c−1 be such that u(x) < a for x ∈ B and u(x) = 0 for x B. Then E exp log(1 − u(x)) η(dx) =1+ ∞ (−1)n n! n=1 u(x1 ) · · · u(xn ) αn (d(x1 , . . . , xn )). (4.28) Since u vanishes outside B, we have P := exp log(1 − u(x)) η(dx) = exp log(1 − u(x)) ηB (dx) . Proof Hence we can assume that η(X \ B) = 0. Since α1 (B) = E[η(B)] < ∞, we can also assume that η(B) < ∞. But then we obtain from Exercise 4.6 that ∞ P= (−1)n Pn , n=0 where P0 := 1 and 1 n! Pn := u(x1 ) · · · u(xn ) η(n) (d(x1 , . . . , xn )), and where we note that η(n) = 0 if n > η(X); see (4.7). Exercise 4.9 asks the reader to prove that 2m−1 2m n=0 n=0 (−1)n Pn ≤ P ≤ (−1)n Pn , These inequalities show that k P − n (−1) Pn ≤ Pk , n=0 m ≥ 1. k ≥ 1. (4.29) The Mecke Equation and Factorial Measures 36 It follows that k E[P] − E ≤ E[P ] = 1 n u(x1 ) · · · u(xk ) αk (d(x1 , . . . , xk )), (−1) P n k k! n=0 where we have used the deﬁnition of the factorial moment measures. The last term can be bounded by ak αk (Bk ) ≤ ak ck , k! which tends to zero as k → ∞. This ﬁnishes the proof. Proposition 4.12 Let η and η be point processes on X with the same factorial moment measures αn , n ≥ 1. Assume that there is a sequence Bk ∈ X, k ∈ N, with Bk ↑ X and numbers ck > 0, k ∈ N, such that αn Bnk ≤ n!cnk , k, n ∈ N. (4.30) d Then η = η . Proof By Proposition 2.10 and monotone convergence it is enough to prove that Lη () = Lη () for each bounded ∈ R+ (X) such that there exists a set B ∈ {Bk : k ∈ N} with (x) = 0 for all x B. This puts us into the setting of Lemma 4.11. Let ∈ R+ (X) have the upper bound a > 0. For each t ∈ [0, −(log(1 − c−1 ))/a) we can apply Lemma 4.11 with u := 1 − e−t . This gives us Lη (t) = Lη (t). Since t → Lη (t) is analytic on (0, ∞), we obtain Lη (t) = Lη (t) for all t ≥ 0 and, in particular, Lη () = Lη (). 4.5 Exercises Exercise 4.1 Let η be a Poisson process on X with intensity measure λ and let A ∈ N have P(η ∈ A) = 0. Use the Mecke equation to show that P(η + δ x ∈ A) = 0 for λa.e. x. Exercise 4.2 Let μ ∈ N be given by (4.3) and let m ∈ N. Show that m−1 m−2 μ(m) (C) = 1C (x1 , . . . , xm ) μ − δ x j (dxm ) μ − δ x j (dxm−1 ) j=1 · · · (μ − δ x1 )(dx2 ) μ(dx1 ), j=1 C∈X . m (4.31) This formula involves integrals with respect to signed measures of the form μ − ν, where μ, ν ∈ N and ν is ﬁnite. These integrals are deﬁned as a diﬀerence of integrals in the natural way. 4.5 Exercises 37 Exercise 4.3 Let μ ∈ N and x ∈ X. Show for all m ∈ N that 1{(x, x1 , . . . , xm ) ∈ ·} + · · · + 1{(x1 , . . . , xm , x) ∈ ·} μ(m) (d(x1 , . . . , xm )) + μ(m+1) = (μ + δ x )(m+1) . (Hint: Use Proposition A.18 to reduce to the case μ(X) < ∞ and then Lemma A.15 to reduce further to the case (4.3) with k ∈ N.) Exercise 4.4 Let μ ∈ N. Use the recursion (4.9) to show that (4.6), (4.7) and (4.8) hold. Exercise 4.5 Let μ ∈ N be given by μ := kj=1 δ x j for some k ∈ N0 and some x1 , . . . , xk ∈ X. Let u : X → R be measurable. Show that k k (−1)n u(x1 ) · · · u(xn ) μ(n) (d(x1 , . . . , xn )). (1 − u(x j )) = 1 + n! j=1 n=1 Exercise 4.6 Let μ ∈ N such that μ(X) < ∞ and let u ∈ R+ (X) satisfy u < 1. Show that exp log(1 − u(x)) μ(dx) n ∞ (−1)n =1+ u(x j ) μ(n) (d(x1 , . . . , xn )). n! n=1 j=1 (Hint: If u takes only a ﬁnite number of values, then the result follows from Lemma A.15 and Exercise 4.5.) Exercise 4.7 (Converse to Theorem 4.4) Let m ∈ N with m > 1. Prove or disprove that for any σﬁnite measure space (X, X, λ), if η is a point process on X satisfying (4.11) for all f ∈ R+ (Xm × N), then η is a Poisson process with intensity measure λ. (For m = 1, this is true by Theorem 4.1.) Exercise 4.8 Give another (inductive) proof of the multivariate Mecke identity (4.11) using the univariate version (4.2) and the recursion (4.9). Exercise 4.9 Prove the inequalities (4.29). (Hint: Use induction.) Exercise 4.10 Let η be a Poisson process on X with intensity measure λ and let B ∈ X with 0 < λ(B) < ∞. Let U1 , . . . , Un be independent random elements of X with distribution λ(B)−1 λ(B∩·) and assume that (U1 , . . . , Un ) and η are independent. Show that the distribution of η + δU1 + · · · + δUn is absolutely continuous with respect to P(η ∈ ·) and that μ → λ(B)−n μ(n) (Bn ) is a version of the density. 5 Mappings, Markings and Thinnings It was shown in Chapter 3 that an independent superposition of Poisson processes is again Poisson. The properties of a Poisson process are also preserved under other operations. A mapping from the state space to another space induces a Poisson process on the new state space. A more intriguing persistence property is the Poisson nature of positiondependent markings and thinnings of a Poisson process. 5.1 Mappings and Restrictions Consider two measurable spaces (X, X) and (Y, Y) along with a measurable mapping T : X → Y. For any measure μ on (X, X) we deﬁne the image of μ under T (also known as the pushforward of μ), to be the measure T (μ) deﬁned by T (μ) = μ ◦ T −1 , i.e. T (μ)(C) := μ(T −1C), C ∈ Y. (5.1) In particular, if η is a point process on X, then for any ω ∈ Ω, T (η(ω)) is a measure on Y given by T (η(ω))(C) = η(ω, T −1 (C)), C ∈ Y. (5.2) κ n=1 δXn as in (2.4), the If η is a proper point process, i.e. one given by η = deﬁnition of T (η) implies that T (η) = κ δT (Xn ) . (5.3) n=1 Theorem 5.1 (Mapping theorem) Let η be a point process on X with intensity measure λ and let T : X → Y be measurable. Then T (η) is a point process with intensity measure T (λ). If η is a Poisson process, then T (η) is a Poisson process too. 38 5.2 The Marking Theorem 39 Proof We ﬁrst note that T (μ) ∈ N for any μ ∈ N. Indeed, if μ = ∞j=1 μ j , then T (μ) = ∞j=1 T (μ j ). Moreover, if the μ j are N0 valued, so are the T (μ j ). For any C ∈ Y, T (η)(C) is a random variable and by the deﬁnition of the intensity measure its expectation is E[T (η)(C)] = E[η(T −1C)] = λ(T −1C) = T (λ)(C). (5.4) If η is a Poisson process, then it can be checked directly that T (η) is completely independent (property (ii) of Deﬁnition 3.1), and that T (η)(C) has a Poisson distribution with parameter T (λ)(C) (property (i) of Deﬁnition 3.1). If η is a Poisson process on X then we may discard all of its points outside a set B ∈ X to obtain another Poisson process. Recall from (4.16) the deﬁnition of the restriction νB of a measure ν on X to a set B ∈ X. Theorem 5.2 (Restriction theorem) Let η be a Poisson process on X with sﬁnite intensity measure λ and let C1 , C2 , . . . ∈ X be pairwise disjoint. Then ηC1 , ηC2 , . . . are independent Poisson processes with intensity measures λC1 , λC2 , . . . , respectively. Proof As in the proof of Proposition 3.5, it is no restriction of generality to assume that the union of the sets Ci is all of X. (If not, add the complement of this union to the sequence (Ci ).) First note that, for each i ∈ N, ηCi has intensity measure λCi and satisﬁes the two deﬁning properties of a Poisson process. By the existence theorem (Theorem 3.6) we can ﬁnd a sequence ηi , i ∈ N, of independent Poisson processes on a suitable (product) probability space, with ηi having intensity measure λCi for each i. By the superposition theorem (Theorem 3.3), the point process η := ∞ d i=1 ηi is a Poisson process with intensity measure λ. Then η = η by Proposition 3.2. Hence for any k and any f1 , . . . , fk ∈ R+ (N) we have E k i=1 fi (ηCi ) = E fi (ηC i ) = E k i=1 k fi (ηi ) = i=1 k E[ fi (ηi )]. i=1 d Taking into account that ηCi = ηi for all i ∈ N (Proposition 3.2), we get the result. 5.2 The Marking Theorem Suppose that η is a proper point process, i.e. one that can be represented as in (2.4). Suppose that one wishes to give each of the points Xn a random 40 Mappings, Markings and Thinnings mark Yn with values in some measurable space (Y, Y), called the mark space. Given η, these marks are assumed to be independent, while their conditional distribution is allowed to depend on the value of Xn but not on any other information contained in η. This marking procedure yields a point process ξ on the product space X × Y. Theorem 5.6 will show the remarkable fact that if η is a Poisson process then so is ξ. To make the above marking idea precise, let K be a probability kernel from X to Y, that is a mapping K : X × Y → [0, 1] such that K(x, ·) is a probability measure for each x ∈ X and K(·, C) is measurable for each C ∈ Y. Deﬁnition 5.3 Let η = κn=1 δXn be a proper point process on X. Let K be a probability kernel from X to Y. Let Y1 , Y2 , . . . be random elements in Y and assume that the conditional distribution of (Yn )n≤m given κ = m ∈ N and (Xn )n≤m is that of independent random variables with distributions K(Xn , ·), n ≤ m. Then the point process ξ := κ δ(Xn ,Yn ) (5.5) n=1 is called a Kmarking of η. If there is a probability measure Q on Y such that K(x, ·) = Q for all x ∈ X, then ξ is called an independent Qmarking of η. For the rest of this section we ﬁx a probability kernel K from X to Y. If the random variables Yn , n ∈ N, in Deﬁnition 5.3 exist, then we say that the underlying probability space (Ω, F , P) supports a Kmarking of η. We now explain how (Ω, F , P) can be modiﬁed so as to support a marking. Let Ω̃ := Ω × Y∞ be equipped with the product σﬁeld. Deﬁne a probability kernel K̃ from Ω to Y∞ by taking the inﬁnite product K̃(ω, ·) := ∞ # K(Xn (ω), ·), ω ∈ Ω. n=1 We denote a generic element of Y∞ by y = (yn )n≥1 . Then P̃ := 1{(ω, y) ∈ ·} K̃(ω, dy) P(dω) (5.6) is a probability measure on Ω̃ that can be used to describe a Kmarking of η. Indeed, for ω̃ = (ω, y) ∈ Ω̃ we can deﬁne η̃(ω̃) := η(ω) and, for n ∈ N, (X̃n (ω̃), Yn (ω̃)) := (Xn (ω), yn ). Then the distribution of (η̃(X), (X̃n )) under P̃ coincides with that of (η(X), (Xn )) under P. Moreover, it is easy to check that under P̃ the conditional distribution of (Yn )n≤m given η̃(X) = m ∈ N and 5.2 The Marking Theorem 41 (X̃n )n≤m is that of independent random variables with distributions K(X̃n , ·), n ≤ m. This construction is known as an extension of a given probability space so as to support further random elements with a given conditional distribution. In particular, it is no restriction of generality to assume that our ﬁxed probability space supports a Kmarking of η. The next proposition shows among other things that the distribution of a Kmarking of η is uniquely determined by K and the distribution of η. Proposition 5.4 Let ξ be a Kmarking of a proper point process η on X as in Deﬁnition 5.3. Then the Laplace functional of ξ is given by Lξ (u) = Lη (u∗ ), where u∗ (x) := − log u ∈ R+ (X × Y), e−u(x,y) K(x, dy) , (5.7) x ∈ X. (5.8) Recall that N0 := N0 ∪ {∞}. For u ∈ R+ (X × Y) we have that m E 1{κ = m} exp − u(Xk , Yk ) Lξ (u) = Proof k=1 m∈N0 m = E 1{κ = m} exp − u(Xk , yk ) k=1 m∈N0 m K(Xk , dyk ) , k=1 where in the case m = 0 empty sums are set to 0 while empty products are set to 1. Therefore m exp[−u(Xk , yk )]K(Xk , dyk ) . E 1{κ = m} Lξ (u) = k=1 m∈N0 ∗ Using the function u deﬁned by (5.8) this means that m ∗ E 1{κ = m} exp[−u (xk )] Lξ (u) = m∈N0 k=1 m ∗ = E 1{κ = m} exp − u (Xk ) , m∈N0 k=1 which is the righthand side of the asserted identity (5.7). The next result says that the intensity measure of a Kmarking of a point process with intensity measure λ is given by λ ⊗ K, where (5.9) (λ ⊗ K)(C) := 1C (x, y) K(x, dy) λ(dx), C ∈ X ⊗ Y. 42 Mappings, Markings and Thinnings In the case of an independent Qmarking this is the product measure λ ⊗ Q. If λ and K are sﬁnite, then so is λ ⊗ K. Proposition 5.5 Let η be a proper point process on X with intensity measure λ and let ξ be a Kmarking of η. Then ξ is a point process on X × Y with intensity measure λ ⊗ K. Proof that Let C ∈ X ⊗ Y. Similarly to the proof of Proposition 5.4 we have m E 1{κ = m} 1{(Xk , Yk ) ∈ C} E[ξ(C)] = m∈N0 k=1 m∈N0 k=1 m = 1{(Xk , yk ) ∈ C} K(Xk , dyk ) . E 1{κ = m} Using Campbell’s formula (Proposition 2.7) with u ∈ R+ (X) deﬁned by u(x) := 1{(x, y) ∈ C} K(x, dy), x ∈ X, we obtain the result. Now we formulate the previously announced behaviour of Poisson processes under marking. Theorem 5.6 (Marking theorem) Let ξ be a Kmarking of a proper Poisson process η with sﬁnite intensity measure λ. Then ξ is a Poisson process with intensity measure λ ⊗ K. Proof Let u ∈ R+ (X × Y). By Proposition 5.4 and Theorem 3.9, ∗ Lξ (u) = exp − (1 − e−u (x) ) λ(dx) = exp − (1 − e−u(x,y) ) K(x, dy) λ(dx) . Another application of Theorem 3.9 shows that ξ is a Poisson process. Under some technical assumptions we shall see in Proposition 6.16 that any Poisson process on a product space is a Kmarking for some kernel K, determined by the intensity measure. 5.3 Thinnings A thinning keeps the points of a point process η with a probability that may depend on the location and removes them otherwise. Given η, the thinning decisions are independent for diﬀerent points. The formal deﬁnition can be based on a special Kmarking: 5.3 Thinnings 43 Deﬁnition 5.7 Let p : X → [0, 1] be measurable and consider the probability kernel K from X to {0, 1} deﬁned by K p (x, ·) := (1 − p(x))δ0 + p(x)δ1 , x ∈ X. If ξ is a K p marking of a proper point process η, then ξ(· × {1}) is called a pthinning of η. 1 0 0 1 We shall use this terminology also in the case where p(x) ≡ p does not depend on x ∈ X. X X Figure 5.1 Illustration of a marking and a thinning, both based on the same set of marked points. The points on the horizontal axis represent the original point process in the ﬁrst diagram, and the thinned point process in the second diagram. More generally, let pi , i ∈ N, be a sequence of measurable functions from X to [0, 1] such that ∞ pi (x) = 1, x ∈ X. (5.10) i=1 Deﬁne a probability kernel K from X to N by K(x, {i}) := pi (x), x ∈ X, i ∈ N. (5.11) If ξ is a Kmarking of a point process η, then ηi := ξ(· × {i}) is a pi thinning of η for every i ∈ N. By Proposition 5.5, ηi has intensity measure pi (x) λ(dx), where λ is the intensity measure of η. The following generalisation of Proposition 1.3 is consistent with the superposition theorem (Theorem 3.3). Theorem 5.8 Let ξ be a Kmarking of a proper Poisson process η, where K is given as in (5.11). Then ηi := ξ(· × {i}), i ∈ N, are independent Poisson processes. 44 Mappings, Markings and Thinnings Proof By Theorem 5.6, ξ is a Poisson process. Hence we can apply The orem 5.2 with Ci := X × {i} to obtain the result. If η p is a pthinning of a proper point process η then (according to Deﬁnitions 2.4 and 5.7) there is an A ∈ F such that P(A) = 1 and η p (ω) ≤ η(ω) for each ω ∈ A. We can then deﬁne a proper point process η − η p by setting (η − η p )(ω) := η(ω) − η p (ω) for ω ∈ A and (η − η p )(ω) := 0, otherwise. Corollary 5.9 (Thinning theorem) Let p : X → [0, 1] be measurable and let η p be a pthinning of a proper Poisson process η. Then η p and η − η p are independent Poisson processes. 5.4 Exercises Exercise 5.1 (Displacement theorem) Let λ be an sﬁnite measure on the Euclidean space Rd , let Q be a probability measure on Rd and let the convolution λ ∗ Q be the measure on Rd , deﬁned by (λ ∗ Q)(B) := 1B (x + y) λ(dx) Q(dy), B ∈ B(Rd ). Show that λ ∗ Q is sﬁnite. Let η = κn=1 δXn be a Poisson process with intensity measure λ and let (Yn ) be a sequence of independent random vectors with distribution Q that is independent of η. Show that η := κn=1 δXn +Yn is a Poisson process with intensity measure λ ∗ Q. Exercise 5.2 Let η1 and η2 be independent Poisson processes with intensity measures λ1 and λ2 , respectively. Let p be a Radon–Nikodým derivative of λ1 with respect to λ := λ1 +λ2 . Show that η1 has the same distribution as a pthinning of η1 + η2 . Exercise 5.3 Let ξ1 , . . . , ξn be identically distributed point processes and let ξ(n) be an n−1 thinning of ξ := ξ1 + · · · + ξn . Show that ξ(n) has the same intensity measure as ξ1 . Give examples where ξ1 , . . . , ξn are independent and where ξ(n) and ξ1 have (resp. do not have) the same distribution. Exercise 5.4 Let p : X → [0, 1] be measurable and let η p be a pthinning of a proper point process η. Using Proposition 5.4 or otherwise, show that Lη p (u) = E exp log 1 − p(x) + p(x)e−u(x) η(dx) , u ∈ R+ (X). Exercise 5.5 Let η be a proper Poisson process on X with σﬁnite intensity measure λ. Let λ be a σﬁnite measure on X and let ρ := λ + λ . Let h := dλ/dρ (resp. h := dλ /dρ) be the Radon–Nikodým derivative of 5.4 Exercises 45 λ (resp. λ ) with respect to ρ; see Theorem A.10. Let B := {h > h } and deﬁne p : X → [0, 1] by p(x) := h (x)/h(x) for x ∈ B and by p(x) := 1, otherwise. Let η be a pthinning of η and let η be a Poisson process with intensity measure 1X\B (x)(h (x) − h(x)) ρ(dx), independent of η . Show that η + η is a Poisson process with intensity measure λ . Exercise 5.6 (Poisson cluster process) Let K be a probability kernel from X to N(X). Let η be a proper Poisson process on X with intensity measure λ and let A ∈ F such that P(A) = 1 and such that (2.4) holds on A. Let ξ be a Kmarking of η and deﬁne a point process χ on X by setting χ(ω, B) := μ(B) ξ(ω, d(x, μ)), B ∈ X, (5.12) for ω ∈ A and χ(ω, ·) := 0, otherwise. Show that χ has intensity measure λ (B) = μ(B) K(x, dμ) λ(dx), B ∈ X. Show also that the Laplace functional of χ is given by Lχ () = exp − (1 − e−μ() ) λ̃(dμ) , ∈ R+ (X), where λ̃ := (5.13) K(x, ·) λ(dx). Exercise 5.7 Let χ be a Poisson cluster process as in Exercise 5.6 and let B ∈ X. Combine Exercise 2.7 and (5.13) to show that P( χ(B) = 0) = exp − 1{μ(B) > 0} λ̃(dμ) . Exercise 5.8 Let χ be as in Exercise 5.6 and let B ∈ X. Show that P( χ(B) < ∞) = 1 if and only if λ̃({μ ∈ N : μ(B) = ∞}) = 0 and λ̃({μ ∈ N : μ(B) > 0}) < ∞. (Hint: Use P( χ(B) < ∞) = limt↓0 E e−tχ(B) .) Exercise 5.9 Let p ∈ [0, 1) and suppose that η p is a pthinning of a proper point process η. Let f ∈ R+ (X × N) and show that p E f (x, η p + δ x ) (η − η p )(dx) . E f (x, η p ) η p (dx) = 1− p 6 Characterisations of the Poisson Process A point process without multiplicities is said to be simple. For locally ﬁnite simple point processes on a metric space without ﬁxed atoms the two deﬁning properties of a Poisson process are equivalent. In fact, Rényi’s theorem says that in this case even the empty space probabilities suﬃce to imply that the point process is Poisson. On the other hand, a weak (pairwise) version of the complete independence property leads to the same conclusion. A related criterion, based on the factorial moment measures, is also given. 6.1 Borel Spaces In this chapter we assume (X, X) to be a Borel space in the sense of the following deﬁnition. In the ﬁrst section we shall show that a large class of point processes is proper. Deﬁnition 6.1 A Borel space is a measurable space (Y, Y) such that there is a Borelmeasurable bijection ϕ from Y to a Borel subset of the unit interval [0, 1] with measurable inverse. A special case arises when X is a Borel subset of a complete separable metric space (CSMS) and X is the σﬁeld on X generated by the open sets in the inherited metric. In this case, (X, X) is called a Borel subspace of the CSMS; see Section A.2. By Theorem A.19, any Borel subspace X of a CSMS is a Borel space. In particular, X is then a metric space in its own right. Recall that N<∞ (X) denotes the set of all integervalued measures on X. Proposition 6.2 There exist