Markov Random Fields

01.11.1982

Englisch

9780387907086

Springer US

Nein

Y.A. Rozanov

201

Gebundene Ausgabe

Constance M. Elson

Brownian motion, random function, Markov property, Probability distribution, Conditional probability, Probability space, Fields, probability measure, Vector, Zufälliges Feld

PBT - Wahrscheinlichkeitsrechnung und Statistik PBWL - Stochastik

1 General Facts About Probability Distributions.- §1. Probability Spaces.- 1. Measurable Spaces.- 2. Distributions and Measures.- 3. Probability Spaces.- §2. Conditional Distributions.- 1. Conditional Expectation.- 2. Conditional Probability Distributions.- §3. Zero-One Laws. Regularity.- 1. Zero-One Law.- 2. Decomposition Into Regular Components.- §4. Consistent Conditional Distributions.- 1. Consistent Conditional Distributions for a Given Probability Measure.- 2. Probability Measures with Given Conditional Distributions.- 3. Construction of Consistent Conditional Distributions.- §5. Gaussian Probability Distributions.- 1. Basic Definitions and Examples.- 2. Some Useful Propositions.- 3. Gaussian Linear Functionals on Countably-Normed Hilbert Spaces.- 4. Polynomials of Gaussian Variables and Their Conditional Expectations.- 5. Hermite Polynomials and Multiple Stochastic Integrals.- 2 Markov Random Fields.- §1. Basic Definitions and Useful Propositions.- 1. Splitting ?-algebras.- 2. Markov Random Processes.- 3. Random Fields; Markov Property.- 4. Transformations of Distributions which Preserve the Markov Property. Additive Functionals.- §2. Stopping ?-algebras. Random Sets and the Strong Markov Property.- 1. Stopping ?-algebras.- 2. Random Sets.- 3. Compatible Random Sets.- 4. Strong Markov Property.- §3. Gaussian Fields. Markov Behavior in the Wide Sense.- 1. Gaussian Random Fields.- 2. Splitting Spaces.- 3. Markov Property.- 4. Orthogonal Random Fields.- 5. Dual Fields. A Markov Criterion.- 6. Regularity Condition. Decomposition of a Markov Field into Regular and Singular Components.- 3 The Markov Property for Generalized Random Functions.- §1. Biorthogonal Generalized Functions and the Duality Property.- 1. The Meaning of Biorthogonality for Generalized Functions in Hilbert Space.- 2. Duality of Biorthogonal Functions.- 3. The Markov Property for Generalized Functions.- §2. Stationary Generalized Functions.- 1. Spectral Representation of Coupled Stationary Generalized Functions.- 2. Biorthogonal Stationary Functions.- 3. The Duality Condition and a Markov Criterion.- §3. Biorthogonal Generalized Functions Given by a Differential Form.- 1. Basic Definitions.- 2. Conditions for Markov Behavior.- §4. Markov Random Functions Generated by Elliptic Differential Forms.- 1. Levy Brownian Motion.- 2. Structure of Spaces for Given Elliptic Forms.- 3. Boundary Conditions.- 4. Regularity and the Dirichlet Problem.- §5. Stochastic Differential Equations.- 1. Markov Transformations of “White Noise”.- 2. The Interpolation and Extrapolation Problems.- 3. The Brownian Sheet.- 4 Vector-Valued Stationary Functions.- §1. Conditions for Existence of the Dual Field.- 1. Spectral Properties.- 2. Duality.- §2. The Markov Property for Stationary Functions.- 1. The Markov Property When a Dual Field Exists.- 2. Analytic Markov Conditions.- §3. Markov Extensions of Random Processes.- 1. Minimal Nonanticipating Extension.- 2. Markov Stationary Processes.- 3. Stationary Processes with Symmetric Spectra.- Notes.

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