Point ProcessesThere has been much recent research on the theory of point processes, i.e., on random systems consisting of point events occurring in space or time. Applications range from emissions from a radioactive source, occurrences of accidents or machine breakdowns, or of electrical impluses along nerve fibres, to repetitive point events in an individual's medical or social history. Sometimes the point events occur in space rather than time and the application here raneg from statistical physics to geography. The object of this book is to develop the applied mathemathics of point processes at a level which will make the ideas accessible both to the research worker and the postgraduate student in probability and statistics and also to the mathemathically inclined individual in another field interested in using ideas and results. A thorough knowledge of the key notions of elementary probability theory is required to understand the book, but specialised "pure mathematical" coniderations have been avoided. |
Contents
Special models | 3 |
CONTENTS | 5 |
Theoretical framework | 21 |
58 | 80 |
65 | 86 |
Operations on point processes | 97 |
Bibliographic notes 4 | 114 |
ii Doubly stochastic cluster and linear | 123 |
Bibliographic notes 5 | 140 |
ii A Markov construction | 150 |
Bibliographic notes 2 41 | 169 |
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Common terms and phrases
asymptotic autocovariance bivariate class 2 points cluster centres complete intensity function conditional intensity function consider constant counter counting measure counts covariance defined denote density function density g dependent discussion disjoint sets distribution function distribution of mean doubly stochastic Poisson equation example exponential distribution finite follows forward recurrence function G Gaussian process given independent and identically infinitely divisible instant interval distribution interval sequence intervals between successive joint distribution Laplace transform Markov process multiple occurrences multivariate number of points obtained ordinary renewal process origin p₁ parameter particular point process points occur Poisson distribution probability density probability generating function process of rate random variables renewal process s₁ second-order properties semi-Markov process simple spatial process specification stationary process stochastic Poisson process stochastic process studied successive points superposition Suppose survivor function t₁ time-dependent Poisson process u₁ u₂ univariate upcrossings variance X₁ zero