Poisson ProcessesIn 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. |
Contents
Stochastic models for random sets of points | 1 |
Poisson processes in general spaces | 11 |
Sums over Poisson processes | 29 |
Poisson processes on the line | 38 |
Marked Poisson processes | 55 |
Cox processes | 65 |
Stochastic geometry | 73 |
Completely random measures | 79 |
The PoissonDirichlet distribution | 90 |
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Common terms and phrases
A₁ applied argument B₁ Bernoulli process Borel-Cantelli lemma bounded sets Campbell's Theorem Colouring completely random measure constant rate converges countable set Cox processes defined denotes dimension Dirichlet distribution disjoint distribution P(µ equation finite number finite on bounded fixed atoms form a Poisson function f generalised geometrical Hence homogeneous Poisson process II(t independent Poisson processes independent random variables infinite instance integral interval joint distributions k-cubes Kingman law of large Lebesgue measure lemma Mapping Theorem mean measure measurable function measurable sets measure µ non-atomic measure number of points Pn(t Poisson distribution Poisson line process positive probability density process with mean process with rate proof prove queue random countable subset random set random subset rate function Rényi's result Section 2.1 sequence step functions Superposition Theorem Suppose Theorem Let theory vector X₁ Y₁ Y₂ zero αμ ΧΕΠ