The Theory of Stochastic Processes (Grundlehren der Mathematischen Wissenschaften)

Gikhman, Iosif I./ Skorokhod, Anatoli V./ Kotz, S. (TRN)

Springer-Verlag Berlin and Heidelberg GmbH & Co. K（1980/01発売）

• 製本 Hardcover:ハードカバー版／ページ数 582 p.
• 言語 ENG
• 商品コード 9783540065739
• DDC分類 519.2

Full Description

From the Reviews: "Gihman and Skorohod have done an excellent job of presenting the theory in its present state of rich imperfection." --D.W. Stroock, Bulletin of the American Mathematical Society, 1980

Contents

I. Basic Notions of Probability Theory.- 1. Axioms and Definitions.- 2. Independence.- 3. Conditional Probabilities and Conditional Expectations.- 4. Random Functions and Random Mappings.- II. Random Sequences.- 1. Preliminary Remarks.- 2. Semi-Martingales and Martingales.- 3. Series.- 4. Markov Chains.- 5. Markov Chains with a Countable Number of States.- 6. Random Walks on a Lattice.- 7. Local Limit Theorems for Lattice Walks.- 8. Ergodic Theorems.- III. Random Functions.- 1. Some Classes of Random Functions.- 2. Separable Random Functions.- 3. Measurable Random Functions.- 4. A Criterion for the Absence of Discontinuities of the Second Kind.- 5. Continuous Processes.- IV. Linear Theory of Random Processes.- 1. Correlation Functions.- 2. Spectral Representations of Correlation Functions.- 3. A Basic Analysis of Hilbert Random Functions.- 4. Stochastic Measures and Integrals.- 5. Integral Representation of Random Functions.- 6. Linear Transformations.- 7. Physically Realizable Filters.- 8. Forecasting and Filtering of Stationary Processes.- 9. General Theorems on Forecasting Stationary Processes.- V. Probability Measures on Functional Spaces.- 1. Measures Associated with Random Processes.- 2. Measures in Metric Spaces.- 3. Measures on Linear Spaces. Characteristic Functionals.- 4. Measures in ?p Spaces.- 5. Measures in Hilbert Spaces.- 6. Gaussian Measures in a Hilbert Space.- VI. Limit Theorems for Random Processes.- 1. Weak Convergences of Measures in Metric Spaces.- 2. Conditions for Weak Convergence of Measures in Hilbert Spaces.- 3. Sums of Independent Random Variables with Values in a Hilbert Space.- 4. Limit Theorems for Continuous Random Processes.- 5. Limit Theorems for Processes without Discontinuities of the Second Kind.- VII. Absolute Continuity of Measures Associated with Random Processes.- 1. General Theorems on Absolute Continuity.- 2. Admissible Shifts in Hilbert Spaces.- 3. Absolute Continuity of Measures under Mappings of Spaces.- 4. Absolute Continuity of Gaussian Measures in a Hilbert Space.- 5. Equivalence and Orthogonality of Measures Associated with Stationary Gaussian Processes.- 6. General Properties of Densities of Measures Associated with Markov Processes.- VIII. Measurable Functions on Hilbert Spaces.- 1. Measurable Linear Functionals and Operators on Hilbert Spaces.- 2. Measurable Polynomial Functions. Orthogonal Polynomials.- 3. Measurable Mappings.- 4. Calculation of Certain Characteristics of Transformed Measures.- Historical and Bibliographical Remarks.- Corrections.