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Book review

Author(s) König, D.
Schmidt, V.
Title Zufällige Punktprozesse; eine Einführung mit Anwendungsbeispielen
Publisher Teubner
Year of publication 1992
Reviewed by Günter Vojta

In recent years, the theory of stochastic processes and stochastic fields and its applications have experienced an enormous progress. Particularly important is the theory of stochastic point processes. A point process can be defined as a set of isolated points randomly distributed in some space; the points may represent times of events, spatial locations of objects or even paths in phase spaces followed by a stochastic system. Today the applications of point processes range from renewal theory and theory of queues to theories of random walk, of percolation and of transport processes in fractal systems, to quantum field theory and to many other fields of physics, materials research, biology and science generally.

The literature, viz. textbooks or research monographs, on these topics is up to now not very rich. Therefore, texts like the book under review are welcome. This volume offers a good introduction into the whole field of simple and marked point processes in spaces of one and several dimensions together with selected applications, particularly the theory of queues, stochastic geometry and stereology.

The volume is divided into thirteen chapters in which, in turn, there are thoroughly treated fundamentals, representations and characteristic properties of point processes including Campbell measures, Palm distributions and generating functionals, stationary processes, Poisson, Cox and cluster processes, ergodic and mixing properties, semi-Markovian processes, marked point processes, martingale methods for point processes, processes in higher-dimensional spaces and, repeatedly, applications as mentioned. A list of symbols, important references (6 pp.) and an index supplement the text.

The approach chosen is based on counting measures; from there other important representations of point processes follow easily. The book is vividly written, mathematical rigour is aimed at corresponding to the introductory level of the text. Proofs are given for the most part in detail and completely. A larger number of problems (without solutions) are added after each chapter. Unfortunately, spectral representations of point processes and questions of statistical inference are not considered. Applications of point processes in physics are largely outside the scope of the book.

The volume is certainly suitable as a text for graduate-level courses and seminars as well as for self-instruction. Necessary prerequisites for the reader are a good knowledge of probability theory based on a measure-theoretic approach and of martingale theory. The book is recommended to researchers, teachers and students of probability theory and any of its applications.