An introduction to the theory of point processes

590 Pages·2007·3.046 MB·English

by  D.J. Daley, David Vere-Jones

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D.J. Daley D. Vere-Jones
An Introduction to the
Theory of Point Processes
Volume II: General Theory and Structure
Second Edition
ABC
D.J.Daley D.Vere-Jones
CentreforMathematicsanditsApplications SchoolofMathematics,Statistics
MathematicalSciencesInstitute andComputingScience
AustralianNationalUniversity VictoriaUniversityofWellington
Canberra.ACT0200,Australia Wellington,NewZealand
[e-mail] [e-mail]
SeriesEditors:
J.Gani P.Jagers
StochasticAnalysisGroup,CMA DepartmentofMathematicalSciences
AustralianNationalUniversity ChalmersUniversityofTechnology
Canberra.ACT0200 andGöteberg(Gothenberg)
Australia SE-41296Göteberg
Sweden
C.C.Heyde T.G.Kurtz
StochasticAnalysisGroup,CMA DepartmentofMathematics
AustralianNationalUniversity UniversityofWisconsin
Canberra,ACT0200 480LincolnDrive
Australia Madison,WI53706
USA
ISBN978-0-387-21337-8 e-ISBN978-0-387-49835-5
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(cid:1)c 2008,1988bytheAppliedProbabilityTrust
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Preface to Volume II, Second Edition
In this second volume, we set out a general framework for the theory of
point processes, starting from their interpretation as random measures. The
material represents a reorganized version of those parts of Chapters 6–14 of
the first edition not already covered in Volume I, together with a significant
amount of new material.
Contrary to our initial expectations, growth in the theoretical aspects of
thesubjecthasatleastmatchedthegrowthinapplications. Muchoftheorig-
inal text has been substantially revised in order to present a more consistent
treatment of marked as well as simple point processes. This applies particu-
larly to the material on stationary processes in Chapter 12, the Palm theory
covered in Chapter 13, and the discussion of martingales and conditional in-
tensities in Chapter 14. Chapter 15, on spatial point processes, has also been
significantly modified and extended. Essentially new sections include Sec-
tions 10.3 and 10.4 on point processes defined by Markov chains and Markov
point processes in space; Sections 12.7 on long-range dependence and 12.8 on
scale invariance and self-similarity; Sections 13.4 on marked point processes
and convergence to equilibrium and 13.6 on fractal dimensions; Sections 14.6
on random time changes and 14.7 on Poisson embedding and convergence to
equilibrium; much of the material in Sections 15.1–15.4 on spatial processes
is substantially new or revised; and some recent material on point maps and
point stationarity has been included in Section 13.3.
As in the first edition, much of the general theory has been developed in
the context of a complete separable metric space (c.s.m.s. throughout this
volume). Criticaltothischoiceofcontextistheexistenceofawell-developed
theoryofmeasuresonmetricspaces,assetout,forexample,inParthasarathy
vii
viii Preface to Volume II, Second Edition
(1967) or Billingsley (1968). We use this theory at two levels. First, we es-
tablish results concerning the space1 M# and N# of realizations of random
X X
measures and point processes, showing that these spaces themselves can be
regarded as c.s.m.s.s, and paying particular attention to sample path prop-
erties such as the existence of atoms. Second, leaning on these results, we
use the same framework to discuss the convergence of random measures and
point processes. The fact that the same theory appears at both levels lends
unity and economy to the development, although care needs to be taken in
discriminating between the two levels.
Thetextofthisvolumenecessarilyassumesgreaterfamiliaritywithaspects
of measure theory and topology than was the case in Volume I, and the first
twoappendicesattheendofVolumeIareaimedathelpingthereaderinthis
regard. The third appendix reviews some of the material from martingale
theory and the general theory of processes that underlies the discussion of
predictability and conditional intensities in Chapter 14.
As was the case in Volume I, we are very much indebted to the friends,
critics, reviewers, and readers who have supplied us with comments, sugges-
tions,andcorrectionsatvariousstagesinthepreparationofthisvolume. The
list is too long to include in full, but we would like to mention in particular
thecontinuingsupportandadvicewehavehadfromRobinMilne,ValIsham,
RickSchoenberg,GuntherLast,andourlong-sufferingcolleaguesinCanberra
and Wellington. The patience and expertise of Springer Verlag, as mediated
through our long-continued contacts with John Kimmel, are also very much
appreciated.
Daryl Daley David Vere-Jones
Canberra, Australia Wellington, New Zealand
1In this edition we use M# (and N#) to denote spaces of boundedly finite (counting)
X X
measuresonX whereinthefirsteditionweusedM(cid:0)X (andN(cid:1)X),respectively.
Contents
Preface to Volume II, Second Edition vii
Principal Notation xii
Concordance of Statements from the First Edition xvi
9 Basic Theory of Random Measures and Point Processes 1
9.1 Definitions and Examples 2
9.2 Finite-Dimensional Distributions and the Existence Theorem 25
9.3 Sample Path Properties: Atoms and Orderliness 38
9.4 Functionals: Definitions and Basic Properties 52
9.5 Moment Measures and Expansions of Functionals 65
10 Special Classes of Processes 76
10.1 Completely Random Measures 77
10.2 Infinitely Divisible Point Processes 87
10.3 Point Processes Defined by Markov Chains 95
10.4 Markov Point Processes 118
11 Convergence Concepts and Limit Theorems 131
11.1 Modes of Convergence for Random Measures
and Point Processes 132
11.2 Limit Theorems for Superpositions 146
11.3 Thinned Point Processes 155
11.4 Random Translations 166
ix
x Contents
12 Stationary Point Processes and Random Measures 176
12.1 Stationarity: Basic Concepts 177
12.2 Ergodic Theorems 194
12.3 Mixing Conditions 206
12.4 Stationary Infinitely Divisible Point Processes 216
12.5 Asymptotic Stationarity and Convergence to Equilibrium 222
12.6 Moment Stationarity and Higher-order Ergodic Theorems 236
12.7 Long-range Dependence 249
12.8 Scale-invariance and Self-similarity 255
13 Palm Theory 268
13.1 Campbell Measures and Palm Distributions 269
13.2 Palm Theory for Stationary Random Measures 284
13.3 Interval- and Point-stationarity 299
13.4 Marked Point Processes, Ergodic Theorems,
and Convergence to Equilibrium 317
13.5 Cluster Iterates 334
13.6 Fractal Dimensions 340
14 Evolutionary Processes and Predictability 355
14.1 Compensators and Martingales 356
14.2 Campbell Measure and Predictability 376
14.3 Conditional Intensities 390
14.4 Filters and Likelihood Ratios 400
14.5 A Central Limit Theorem 412
14.6 Random Time Change 418
14.7 Poisson Embedding and Existence Theorems 426
14.8 Point Process Entropy and a Shannon–MacMillan Theorem 440
15 Spatial Point Processes 457
15.1 Descriptive Aspects: Distance Properties 458
15.2 Directional Properties and Isotropy 466
15.3 Stationary Line Processes in the Plane 471
15.4 Space–Time Processes 485
15.5 The Papangelou Intensity and Finite Point Patterns 506
15.6 Modified Campbell Measures and Papangelou Kernels 518
15.7 The Papangelou Intensity Measure and Exvisibility 526
References with Index 537
Subject Index 557