Statistical Analysis of Random Fields

Inhaltsverzeichnis

• 1. Elements of the Theory of Random Fields.
• 1.1 Basic concepts and notation.
• 1.2 Homogeneous and isotropic random fields.
• 1.3 Spectral properties of higher order moments of random fields.
• 1.4 Some properties of the uniform distribution.
• 1.5 Variances of integrals of random fields.
• 1.6 Weak dependence conditions for random fields.
• 1.7 A central limit theorem.
• 1.8 Moment inequalities.
• 1.9 Invariance principle.
• 2. Limit Theorems for Functionals of Gaussian Fields.
• 2.1 Variances of integrals of local Gaussian functionals.
• 2.2 Reduction conditions for strongly dependent random fields.
• 2.3 Central limit theorem for non-linear transformations of Gaussian fields.
• 2.4 Approximation for distribution of geometric functional of Gaussian fields.
• 2.5 Reduction conditions for weighted functionals.
• 2.6 Reduction conditions for functionals depending on a parameter.
• 2.7 Reduction conditions for measures of excess over a moving level.
• 2.8 Reduction conditions for characteristics of the excess over a radial surface.
• 2.9 Multiple stochastic integrals.
• 2.10 Conditions for attraction of functionals of homogeneous isotropic Gaussian fields to semi-stable processes.
• 3. Estimation of Mathematical Expectation.
• 3.1 Asymptotic properties of the least squares estimators for linear regression coefficients.
• 3.2 Consistency of the least squares estimate under non-linear parametrization.
• 3.3 Asymptotic expansion of least squares estimators.
• 3.4 Asymptotic normality and convergence of moments for least squares estimators.
• 3.5 Consistency of the least moduli estimators.
• 3.6 Asymptotic normality of the least moduli estimators.
• 4. Estimation of the Correlation Function.
• 4.1 Definition of estimators.
• 4.2 Consistency.
• 4.3 Asymptotic normality.
• 4.4 Asymptotic normality. The caseof a homogeneous isotropic field.
• 4.5 Estimation by means of several independent sample functions.
• 4.6 Confidence intervals.
• References.
• Comments.