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Book review

Author(s) Ivanov, A.V.
Leonenko, N.N.
Title Statistical Analysis of Random Fields
Publisher Kluwer Academic Publishers
Year of publication 1989
Reviewed by Ilie Parpucea

This volume in intended to present an investigation of the basic problems of the statistics of random fields. The specific nature of random fields manifests itself when studying random functions whose properties are coordinated with the algebraic structure of the space in the same manner as the specific nature of random processes (a function of a single variable viewed as time) are revealed in coordination with the ordering structure. It is usually assumed in problems of statistics of random processes that a process (as a rule a single realization of it) is observed on a time interval that extends to infinity. Based on these observations: (1) one constructs estimators of parameters and(2) one tests hypotheses about the distribution of the process or about the form of its basic characteristic on so on.

An analogous approach is utilized in this book for the study of homogeneous isitropic random fields. It is assumed that a field is observed in an expanding domain and based on these observatlons, problems (1) and (2) are solved for random fields. The book devotes major attention to an investigation of limit distributions for various specific functionals of a geometric nature for Gaussian random fields prossessing strongly and weakly decreasing correlations.

Substantial results are obtained in estimation theory of the first two moments of random fields with a continuous parameter, that is, the mathematical expectation and correlation function.

Chapter 1 presents a discussion of random fields, spectral expansions of homogeneous isotropic random fields, the central limit theorem and the invariance principle for mixing random fields under weak dependence. Chapter 2 embraces limit theorems for functionals of Gaussian fields with strong and weak dependedce, and limit theorems for distributions of functionals of geometric type of Gaussian fields. Chapter 3 deals with the statistical estimations of the unknown mean of random fields. Chapter 4 considers various statistical problems dealing with the estimation of the correlation function of a random field.

Both specialists in the theory of random fields and scient.ists working in related areas that utilize this theory will no doubt find a great deal of new and useful information in this book.

The authors of book are well-known researchers in the field of theory of random fields.