Limit Theorems for Random Fields with Singular Spectrum

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 Stochastic 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.