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.
