An Introduction to the Theory of Point Processes: Volume II: General Theory and Structure

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Springer Science & Business Media, Nov 12, 2007 - Mathematics - 573 pages
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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 An Introduction to the Theory of Point Processes in two volumes with subtitles Volume I: Elementary Theory and Methods and Volume II: General Theory and Structure.

Volume I contains the introductory chapters from the first edition together with an account of basic models, second order theory, and an informal account of prediction, with the aim of making the material accessible to readers primarily interested in models and applications. It also has three appendices that review the mathematical background needed mainly in Volume II.

Volume II sets out the basic theory of random measures and point processes in a unified setting and continues with the more theoretical topics of the first edition: limit theorems, ergodic theory, Palm theory, and evolutionary behaviour via martingales and conditional intensity. The very substantial new material in this second volume includes expanded discussions of marked point processes, convergence to equilibrium, and the structure of spatial point processes.

 

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Contents

Basic Theory of Random Measures and Point Processes
1
91 Definitions and Examples
2
92 FiniteDimensional Distributions and the Existence Theorem
25
Atoms and Orderliness
38
Definitions and Basic Properties
52
95 Moment Measures and Expansions of Functionals
65
Special Classes of Processes
76
101 Completely Random Measures
77
131 Campbell Measures and Palm Distributions
269
132 Palm Theory for Stationary Random Measures
284
133 Interval and Pointstationarity
299
134 Marked Point Processes Ergodic Theorems and Convergence to Equilibrium
317
135 Cluster Iterates
334
136 Fractal Dimensions
340
Evolutionary Processes and Predictability
355
141 Compensators and Martingales
356

102 Infinitely Divisible Point Processes
87
103 Point Processes Defined by Markov Chains
95
104 Markov Point Processes
118
Convergence Concepts and Limit Theorems
131
111 Modes of Convergence for Random Measures and Point Processes
132
112 Limit Theorems for Superpositions
146
113 Thinned Point Processes
155
114 Random Translations
166
Stationary Point Processes and Random Measures
176
Basic Concepts
177
122 Ergodic Theorems
194
123 Mixing Conditions
206
124 Stationary Infinitely Divisible Point Processes
216
and Convergence to Equilibrium
222
and Higherorder Ergodic Theorems
236
127 Longrange Dependence
249
128 Scaleinvariance and Selfsimilarity
255
Palm Theory
268
142 Campbell Measure and Predictability
376
143 Conditional Intensities
390
144 Filters and Likelihood Ratios
400
145 A Central Limit Theorem
412
146 Random Time Change
418
147 Poisson Embedding and Existence Theorems
426
and a ShannonMacMillan Theorem
440
Spatial Point Processes
457
Distance Properties
458
152 Directional Properties and Isotropy
466
153 Stationary Line Processes in the Plane
471
154 SpaceTime Processes
485
155 The Papangelou Intensity and Finite Point Patterns
506
156 Modified Campbell Measures and Papangelou Kernels
518
157 The Papangelou Intensity Measure and Exvisibility
526
References with Index
537
Subject Index
557
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Page 549 - Point Processes," in Stochastic Point Processes, PAW Lewis (Ed.), Wiley, New York, 1972, 1-54. AND GS SHEDLER, "Statistical Analysis of Non-stationary Series of Events in a Data Base System,
Page 539 - DR (1972). The spectral analysis of stationary interval functions. Proc. Sixth Berkeley Symp. Math. Statist.
Page 550 - Moyal, JE (1962). The general theory of stochastic population processes. Acta Math.. 108, 1-31.
Page 550 - Bridging the gap between a stationary point process and its Palm distribution, Statistica Neerlandica 48, 1 (1994), 37-62.
Page 540 - Statist. .39, 1007-1019. (1971) Weakly stationary point processes and random measures. J. Roy. Statist. Soc. Ser. B 33, 406-428.
Page 549 - Bemerkungen zu einer Arbeit von Nguyen Xuan Xanh und Hans Zessin. Math. Nachr. 88, 117-127.

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