Limit Theorems for Random Fields with Singular Spectrum / Edition 1

by Nicolai Leonenko

Overview

This book 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 2 is devoted to limit theorems for spherical averages of nonlinear transformations of Gaussian and chi-square random fields. Chapter 3 summarises 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 4. Lastly, Chapter 5 deals with some problems for statistical analysis of random fields with singular spectrum.
Audience: This book will be of interest to mathematicians who use random fields in engineering or other applications.

Product Details

ISBN-13:	9780792356356
Publisher:	Springer Netherlands
Publication date:	02/28/1999
Series:	Mathematics and Its Applications , #465
Edition description:	1999
Pages:	406
Product dimensions:	6.10(w) x 9.25(h) x 0.24(d)

Table of Contents

1 Second-Order Analysis of Random Fields.- 1.1 Basic Concepts and Notation.- 1.2 Elements of Spectral Theory of Random Fields.- 1.3 Models of Random Processes and Fields with Singular Spectrum.- 1.4 Tauberian and Abelian Theorems for Correlation Function of Homogeneous Isotropic Random Fields.- 2 Limit Theorems for Non-Linear Transformations of Random Fields.- 2.1 Some Properties of Gaussian and X-Squared Random Fields.- 2.2 Reduction Theorems for the Local Functionals of Random Fields with Slowly Decaying Correlations.- 2.3 Multiple Shastic Integrals.- 2.4 Non-Central Limit Theorems for Local Functionals of Random Fields.- 3 Asymptotic Distributions of Geometric Functionals of Random Fields.- 3.1 Limit Distributions for Characteristics of the Excess above a Level for Gaussian Fields.- 3.2 Limiting Distributions for the Excess Over a Radial Surface of X-Squared Random Fields.- 3.3 Spherical Measures of Excess over of Moving Level.- 3.4 Sojourns of Multi-Dimensional Gaussian Fields with Dependent Components.- 3.5 Asymptotic Normality of Random ‘Area of Surface’ of Planar Gaussian Field.- 3.6 Asymptotics for Occupation Densities of Gaussian and X-Squared Random Fields.- 4 Limit Theorems For Solutions of The Burgers’ Equation with Random Data.- 4.1 Physical Motivation and Recent History.- 4.2 Hopf-Cole Solution.- 4.3 Parabolic Asymptotics for Weakly Dependent Random Data: the Gaussian Scenario.- 4.4 Parabolic Limits for Strongly Dependent Random Initial Conditions: the Gaussian Scenario.- 4.5 Parabolic Limits for Strongly Dependent Random Data: the Non-Gaussian Scenario.- 4.6 Exact Parabolic Asymptotics for Singular Burgers’ Equation.- 4.7 Hyperbolic Asymptotics for Rescaled Solutions of Burgers’ Equation.- 5 Statistical Problems for Random Fields withSingular Spectrum.- 5.1 Estimation of Mathematical Expectation.- 5.2 Estimation of the Covariance Function.- 5.3 Efficient Estimation of Regression Coefficients of a Random Fields Observed on the Sphere.- 5.4 Estimation in the Frequency Domain.- Comments.