Limit Theorems for Random Fields with Singular Spectrum

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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.