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Mathematics - Probability Theory and Stochastic Processes | The Theory of Stochastic Processes I

The Theory of Stochastic Processes I

Gikhman, Iosif I., Skorokhod, Anatoli V.

Translated by Kotz, S.

Originally published as Vol. 210 in the series: Grundlehren der mathematischen Wissenschaften

Reprint of the 1st ed. Berlin Heidelberg New York 1974, VIII, 574 p.

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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 in Bulletin of the American Mathematical Society, 1980

"To call this work encyclopedic would not give an accurate picture of its content and style. Some parts read like a textbook, but others are more technical and contain relatively new results. ... The exposition is robust and explicit, as one has come to expect of the Russian tradition of mathematical writing. The set when completed will be an invaluable source of information and reference in this ever-expanding field"
K.L. Chung in American Scientist, 1977

"..., the subject has grown enormously since 1953, and there will never be a true successor to Doob's book, but Gihman and Skorohod's three volumes will, I think, occupy a rather similar position as an invaluable tool of reference for all probability theorists. ... The dominant impression is of the authors' mastery of their material, and of their confident insight into its underlying structure. ..."
J.F.C. Kingman in Bulletin of the London Mathematical Society, 1977

Content Level » Research

Keywords » 60-02 - 60Jxx - 60Kxx - YellowSale2006 - stochastic processes

Related subjects » Probability Theory and Stochastic Processes

Table of 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.

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