An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods
Point processes and random measures find wide applicability in telecommunications, earthquakes, image analysis, spatial point patterns, and stereology, to name but a few areas. The authors have made a major reshaping of their work in their first edition of 1988 and now present their Introduction to the Theory of Point Processes in two volumes with sub-titles Elementary Theory and Models and General Theory and Structure. Volume One contains the introductory chapters from the first edition, together with an informal treatment of some of the later material intended to make it more accessible to readers primarily interested in models and applications. The main new material in this volume relates to marked point processes and to processes evolving in time, where the conditional intensity methodology provides a basis for model building, inference, and prediction. There are abundant examples whose purpose is both didactic and to illustrate further applications of the ideas and models that are the main substance of the text. Volume Two returns to the general theory, with additional material on marked and spatial processes. The necessary mathematical background is reviewed in appendices located in Volume One. Daryl Daley is a Senior Fellow in the Centre for Mathematics and Applications at the Australian National University, with research publications in a diverse range of applied probability models and their analysis; he is co-author with Joe Gani of an introductory text in epidemic modelling. David Vere-Jones is an Emeritus Professor at Victoria University of Wellington, widely known for his contributions to Markov chains, point processes, applications in seismology, and statistical education. He is a fellow and Gold Medallist of the Royal Society of New Zealand, and a director of the consulting group "Statistical Research Associates."
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Basic Properties of the Poisson Process
Simple Results for Stationary Point Processes on the Line
Finite Point Processes
Conditional Intensities and Likelihoods
SecondOrder Properties of Stationary Point Processes
A1 A Review of Some Basic Concepts
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absolutely continuous atom Bartlett spectrum bivariate Borel sets bounded Borel set bounded set boundedly finite Chapter cluster centre compact Complements to Section component compound Poisson compound Poisson process conditional intensity function convergence corresponding covariance measure Cox process defined definition denote discussion equation equivalent Example Exercises and Complements exists exponential extended finite point process follows Fourier transform given ground process Hawkes process hazard function hence independent integral interval Janossy densities Janossy measures Lebesgue measure Lemma likelihood linear measurable function metric space nonnegative p.p.d. measure parameter Poisson process probability measure proof properties Proposition random measure random variables relation renewal process satisfies sequence Show signed measure ſº spectral stationary point stationary point process subsets Suppose symmetric theorem theory topology totally finite Vere-Jones Wold process