## 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. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### 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 |

100 | |

102 | |

103 | |

### Other editions - View all

### Common terms and phrases

applied argument assumption Bernoulli process Borel-Cantelli lemma bounded sets Campbell's Theorem Colouring completely random measure constant rate converges countable set counts N(A Cox processes customers defined denotes dimension Dirichlet distribution equation Ewens sampling formula example fc-cubes finite number finite on bounded fixed atoms form a Poisson function f generalised Hence homogeneous Poisson process homogeneous process independent Poisson processes independent random variables infinite instance integral interval invariant joint distributions large numbers law of large Lebesgue measure Mapping Theorem mean measure measurable sets N(Aj non-atomic measure number of points particular plane Poisson distribution Poisson line process positive probability density process of constant process with mean process with rate product space proof prove queue random countable subset random points random set random subset rate function result Section 2.1 sequence step functions stochastic Superposition Theorem Suppose Theorem Let Theorem shows theory transformations values zero