×

Statistical analysis of random fields. Transl. from the Russian by A. I. Kochubinsky. Transl. ed. by S. Kotz. (English) Zbl 0713.62094

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

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
PDFBibTeX XMLCite