Ivanov, A. V.; Leonenko, N. N. Kotz, S. (ed.) Statistical analysis of random fields. Transl. from the Russian by A. I. Kochubinsky. Transl. ed. by S. Kotz. (English) Zbl 0713.62094 Mathematics and Its Applications: Soviet Series, 28. Dordrecht etc.: Kluwer Academic Publishers. x, 244 p. Dfl 170.00; £53.00; $ 89.00 (1989). The book presents well selected topics from \(L^ 2\)-theory (i.e., Fourier theory), and asymptotic statistical analysis of homogeneous, isotropic random fields assumed to be Gaussian in their good part, under some conditions on dependence and mixing. Fourier analysis of random functions admits a relatively easy extension of most important one- dimensional time theory statements to the multi-parameter time case. Thin martingale and Markov methods are off the subject of this book; for these topics see e.g. J. B. Walsh [Ecole d’été de probabilités de Saint-Flour XIV-1984, Lect. Notes Math. 1180, 265-437 (1986; Zbl 0608.60060)], and the book of Yu. A. Rozanov, Markov random fields. (1982; Zbl 0498.60057). Chapter 1 summarizes preliminary notions, methods and fundamental theorems of \(L^ 2\)-theory. Chapter 2 contains asymptotic upper estimates, strong and weak limit theorems for nonlinear integral functionals, like sample moments, and exceedes those for Gaussian fields, as the sample domain goes to infinity. The weak limit theorems consider mainly the central limits, the white noise one, or its integral, either in functional limit theorems. Chapters 3 and 4 deal with asymptotic parametric and non-parametric estimation of the mean and the correlation functions. Limit theorems on asymptotic normality of some practically interesting estimators, strong convergence of the sample moments and construction of confidence intervals are exposed. The book can be of use to those dealing with random fields theory and statistical applications, and to post-graduate students, as well. Reviewer: E.I.Trofimov Cited in 3 ReviewsCited in 100 Documents MSC: 62M40 Random fields; image analysis 62G20 Asymptotic properties of nonparametric inference 62F12 Asymptotic properties of parametric estimators 62-02 Research exposition (monographs, survey articles) pertaining to statistics 60G60 Random fields 60F17 Functional limit theorems; invariance principles 60F15 Strong limit theorems 60F05 Central limit and other weak theorems 60-02 Research exposition (monographs, survey articles) pertaining to probability theory Keywords:Gaussian random fields; L2-theory; homogeneous, isotropic random fields; Fourier analysis of random functions; asymptotic upper estimates; nonlinear integral functionals; sample moments; white noise; mean; correlation functions; asymptotic normality; strong convergence of the sample moments; construction of confidence intervals Citations:Zbl 0608.60060; Zbl 0498.60057 PDFBibTeX XMLCite \textit{A. V. Ivanov} et al., Statistical analysis of random fields. Transl. from the Russian by A. I. Kochubinsky. Transl. ed. by S. Kotz. Dordrecht etc.: Kluwer Academic Publishers (1989; Zbl 0713.62094)