Gaussian Random Processes - I. A. Ibragimov

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Sommaire:
I Preliminaries.- I.1 Gaussian Probability Distribution in a Euclidean Space.- I.2 Gaussian Random Functions with Prescribed Probability Measure.- I.3 Lemmas on the Convergence of Gaussian Variables.- I.4 Gaussian Variables in a Hilbert Space.- I.5 Conditional Probability Distributions and Conditional Expectations.- I.6 Gaussian Stationary Processes and the Spectral Representation.- II The Structures of the Spaces H(T) and LT(F).- II. 1 Preliminaries.- II.2 The Spaces L+(F) and L-(F).- II.3 The Construction of Spaces LT(F) When T Is a Finite Interval.- II.4 The Projection of L+(F) on L-(F).- II.5 The Structure of the ?-algebra of Events U(T).- III Equivalent Gaussian Distributions and their Densities.- III.1 Preliminaries.- III.2 Some Conditions for Gaussian Measures to be Equivalent.- III.3 General Conditions for Equivalence and Formulas for Density of Equivalent Distributions.- III.4 Further Investigation of Equivalence Conditions.- IV Conditions for Regularity of Stationary Random Processes.- IV.1 Preliminaries.- IV.2 Regularity Conditions and Operators Bt.- IV.3 Conditions for Information Regularity.- IV.4 Conditions for Absolute Regularity and Processes with Discrete Time.- IV.5 Conditions for Absolute Regularity and Processes with Continuous Time.- V Complete Regularity and Processes with Discrete Time.- V.l Definitions and Preliminary Constructions with Examples.- V.2 The First Method of Study: Helson-Sarason's Theorem.- V.3 The Second Method of Study: Local Conditions.- V.4 Local Conditions (continued).- V.5 Corollaries to the Basic Theorems with Examples.- V.6 Intensive Mixing.- VI Complete Regularity and Processes with Continuous Time.- VI.1 Introduction.- VI.2 The Investigation of a Particular Function ?(T;?).- VI.3 The Proof of the Basic Theorem onNecessity.- VI.4 The Behavior of the Spectral Density on the Entire Line.- VI.5 Sufficiency.- VI.6 A Special Class of Stationary Processes.- VII Filtering and Estimation of the Mean.- VII.1 Unbiased Estimates.- VII.2 Estimation of the Mean Value and the Method of Least Squares.- VII.3 Consistent Pseudo-Best Estimates.- VII.4 Estimation of Regression Coefficients.- References.