Stochastic geometry and its applications. 2nd ed. (English) Zbl 0838.60002

Wiley Series in Probability and Mathematical Statistics. Chichester: John Wiley & Sons Ltd. xix, 436 p. (1995).
The book provides a modern and up-to-date treatment of the stochastic geometry mathematical models and appropriate statistical methods for analysis of arbitrary random fields of geometrical objects occurring in many areas of science and technology. Following the style of the first edition (1977; Zbl 0622.60019), the authors make the results and methods of stochastic geometry and spatial statistics more generally accessible to practitioners and non-theoreticians. The book is also ideal as an introduction to the subject for mathematicians. The style and presentation of the second edition is better than that of the first one. The authors present many of the new results, ideas and developments in the field of Boolean models, stereology, random shapes, Gibbs processes, random tessellations and spatial statistics since 1987 and details of the use of the stochastic geometry methods to solve theoretical and physical problems in a wide range of disciplines. The progress of these years is also visible in the jacket of this book. The exposition is mathematically precise and takes into account the latest results. However, in many cases proofs are omitted. The level of exposition is uneven and the subjects are treated with varying thoroughness: some topics are illustrated by numerical examples, some results are stated without much comment, others are accompanied by heuristic arguments and sometimes substantial issues are dismissed with only a few remarks and a few references to the literature. Applied scientists who may not wish to follow the mathematical arguments in detail will still be able to interpret and use the formulae.
The book contains eleven chapters. The first chapter briefly summarizes the mathematical concepts to be employed throughout the book and introduces various notations and conventions. In Chapter 2 the simplest and most important random point-pattern is studied: the Poisson point process. Chapter 3 discusses the Boolean model which is an important and relatively simple example of a random closed set. Chapter 4 presents the general theory of the point processes, which is a development of the special case of the Poisson process. Some special point process models (Cox process, Neyman-Scott process, hard-core process, Gibbs process) and operations on point processes are considered in Chapter 5. Chapter 6 is devoted to the more general theory of random sets. Chapter 7 briefly introduces the important concept of a random measure on \(\mathbb{R}^d\), which arises throughout the subject at a more theoretical level. Chapter 8 introduces the theory of shape and point processes on special representation spaces and describes their use in analysing random processes of geometric objects. Two examples are discussed in some detail: planar line process and process of shapes of triangles. In Chapter 9 interest is focused on the study of systems of fibres and of systems of surfaces or fragments of surfaces, distributed at random on the plane or in space. Various models of random tessellations by convex polygons or polyhedra are described in Chapter 10. The final Chapter 11 is on stereology, which is of great importance in practice and uses results and ideas from all of the preceding discussion.
The theory is illustrated by many examples drawn from different branches of science; actual data in the form of images are presented, and their statistical analysis is discussed. This volume is a first-rate textbook for graduate level seminars, as well as an extremely useful reference for mathematicians, applied scientists, and engineers. As well as being of great interest to statisticians, this treatment of the subject has proved useful to applied scientists working in fields such as geology, biology, microscopy and materials science, and to pure mathematicians working in random geometry. This book is an outstanding monograph written by excellent creative researchers and a very stimulating contribution to further researches in stochastic geometry.
Reviewer: N.Semejko (Kiev)


60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60D05 Geometric probability and stochastic geometry
60G57 Random measures
52A22 Random convex sets and integral geometry (aspects of convex geometry)
62M99 Inference from stochastic processes


Zbl 0622.60019