Limit Theorems for Random Fields with Singular Spectrum: 465 (Mathematics and Its Applications, 465) - Hardcover

Leonenko, Nicolai

Hardcover

ISBN 10: 0792356357 ISBN 13: 9780792356356

Publisher: Springer, 1999

Synopsis

This book is devoted to an investigation of the basic problems of the the­ ory of random fields which are characterized by certain singular properties (e. g., unboundedness, or vanishing) of their spectral densities. These ran­ dom fields are called, the random fields with singular spectrum, long-memory fields, random fields with long-range dependence, fields with slowly decaying correlations or strongly dependent random fields by various authors. This phenomenon has been observed empirically by many scientists long before suitable mathematical models were known. The methods and results differ significantly from the theory of weakly dependent random fields. The first chapter presents basic concepts of the spectral theory of random fields, some examples of random processes and fields with singular spectrum, Tauberian and Abelian theorems for the covariance function of singular ran­ dom fields. In the second chapter limit theorems for non-linear functionals of random fields with singular spectrum are proved. Chapter 3 summarizes some limit theorems for geometric functionals of random fields with long-range dependence. Limit distributions of the solutions of Burgers equation with random data via parabolic and hyperbolic rescaling are presented in chapter 4. And chapter 5 presents some problems of statistical analysis of random fields with singular spectrum. I would like to thank the editor, Michiel Hazewinkel, for his support. I am grateful to the following students and colleagues: 1. Deriev, A. Olenko, K. Rybasov, L. Sakhno, M. Sharapov, A. Sikorskii, M. Silac-BenSic. I would also like to thank V.Anh, O. Barndorff-Nielsen,Yu. Belyaev, P.

About this title

Synopsis

This text presents limit theorems for nonlinear functionals of random fields with singular spectrum on the basis of various asymptotic expansions. The first chapter treats basic concepts of the spectral theory of random fields, some important examples of random processes, and fields with singular spectrum, and Tauberian and Abelian theorems for covariance function of long-memory random fields. Chapter Two is devoted to limit theorems for spherical averages of nonlinear transformations of Gaussian and chi-square random fields, while chapter three summarizes some limit theorems for geometric type functionals of random fields. Limit theorems for the solutions of Burgers' equation with random data via parabolic and hyperbolic rescaling are demonstrated in Chapter Four. Lastly, Chapter Five deals with some problems for statistical analysis of random fields with singular spectrum.

Edition details

• Publisher: Springer
• Publication date: 1999
• Language: English
• ISBN 10: 0792356357
• ISBN 13: 9780792356356
• Binding: Hardcover
• Number of pages: 414