Poisson Processes

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Clarendon Press, Dec 17, 1992 - Mathematics - 112 pages
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In the theory of random processes there are two that are fundamental, and occur over and over again, often in surprising ways. There is a real sense in which the deepest results are concerned with their interplay. One, the Bachelier Wiener model of Brownian motion, has been the subject of many books. The other, the Poisson process, seems at first sight humbler and less worthy of study in its own right. Nearly every book mentions it, but most hurry past to more general point processes or Markov chains. This comparative neglect is ill judged, and stems from a lack of perception of the real importance of the Poisson process. This distortion partly comes about from a restriction to one dimension, while the theory becomes more natural in more general context. This book attempts to redress the balance. It records Kingman's fascination with the beauty and wide applicability of Poisson processes in one or more dimensions. The mathematical theory is powerful, and a few key results often produce surprising consequences.
 

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Contents

Stochastic models for random sets of points
1
12 The Poisson distribution
3
13 Probability spaces for Poisson processes
7
14 The inevitability of the Poisson distribution
9
Poisson processes in general spaces
11
22 The Superposition Theorem
14
23 The Mapping Theorem
17
24 The Bernoulli process
21
56 The orbital motorway
63
Cox processes
65
62 Cox processes in ecology
66
63 The BorelTanner distribution
68
64 Cox processes and renewal processes
71
Stochastic geometry
73
72 Line processes
74
73 Cox line processes
77

25 The Existence Theorem
23
Sums over Poisson processes
25
32 Campbells Theorem
28
33 The characteristic functional
31
34 Renyis Theorem
33
Poisson Processes on the line
38
42 The Law of Large Numbers
41
43 Queues
44
44 Bartletts Theorem
47
45 Nonhomogeneous processes
50
Marked Poisson processes
53
52 The product space representation
55
53 Campbells Theorem revisited
57
54 The wide motorway
59
55 Ecological models
61
Completely random measures
79
82 Construction from Poisson processes
82
83 The Blackwell argument
84
84 Subordinators
87
The PoissonDirichlet distribution
90
92 The Dirichlet process
92
93 The PoissonDirichlet limit
93
94 The Moran subordinator
94
95 The Ewens sampling formula
96
96 Sizebiased sampling
98
References
100
Index of Theorems
102
Subject index
103
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