I An Introduction to Point Processes.- 1 Strong Approximation.- 1.1 Motivation and Basic Concepts.- 1.2 The Poisson Process: Finite Intensity Measure.- 1.3 Poisson and Binomial Distributions.- 1.4 Approximation of Empirical Processes.- E.1 Exercises and Supplements.- 2 Poisson and Cox Processes.- 2.1 ?-Finite Point Processes.- 2.2 Mixtures of Point Processes.- 2.3 Random Measures.- 2.4 Important Operations.- E.2 Exercises and Supplements.- 3 Densities and Distances.- 3.1 Densities of Point Processes.- 3.2 Distances Between Poisson Processes.- E.3 Exercises and Supplements.- II Point Processes in Action.- 4 Nonparametric Curve Estimation.- 4.1 Nonparametric Intensity Estimation.- 4.2 Nonparametric Regression.- E.4 Exercises and Supplements.- 5 Sampling from Finite Populations.- 5.1 Sampling Designs, Sampling Processes.- 5.2 Superpopulation Models.- 5.3 Campbell Theorem: Finite Populations.- E.5 Exercises and Supplements.- 6 Extreme Value Models.- 6.1 Models for Univariate Exceedances.- 6.2 Multivariate Extreme Value Models.- E.6 Exercises and Supplements.- 7 Image Restoration, Spatial Statistics.- 7.1 Inverse Problems, Missing Data.- 7.2 Transformation of Point Processes.- 7.3 Palm Distribution, Campbell Measure.- 7.4 Gibbs Distributions.- 7.5 Line Processes.- 7.6 Spatial Statistics.- E.7 Exercises and Supplements.- III An Outlook on Further Important Approaches.- 8 Weak Approximation.- 8.1 Basic Technical Concepts.- 8.2 Point Processes of Exceedances.- 8.3 The Global Poissonization Technique.- E.8 Exercises and Supplements.- 9 Counting Processes and Martingales.- 9.1 Compensators and Intensity Processes.- E.9 Exercises and Supplements.- Author Index.
Chapters 1 and 2 of this book originated from a manuscript that was dis tributed shortly before aseminar. Each of the participants had acquired a solid background in probability theory and statistics, yet the knowledge of point or counting processes was partly restricted to the homogeneous Poisson counting process buHt by partial sums of exponential random vari ables. These preparations turned out to be successful and, consequently, there was time left during the seminar to study some of the topics outlined in Chapters 3-9. This story reveals the main intention of writing this book, namely, to facilitate first steps into the field of point processes. To avoid misunderstanding, this book is not written in an informal style but in strictly mathematical terms. The book provides extensive explana tions and detailed proofs (including the discussion of measurability prob lems, etc.) which will be helpful to gain insight on the subject. Although this book is of an introductionary nature it includes much new material and presents an elaboration of various topics in a way that cannot be found elsewhere. A broader knowledge and deeper understanding of the subject may be gained by consulting well-known monographs such as those of Daley and Vere-Jones , Resnick , Matthes, Kerstan and Mecke , Karr , Kallenberg , among others, as weH as originalliterature. For that enterprise, introductory remarks about various specific topics and extensive references may serve as a guideline.