Author | Gihman, Iosif Il'ich. author |
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Title | The Theory of Stochastic Processes I [electronic resource] / by Iosif Il'ich Gihman, Anatoliฤญ Vladimirovich Skorokhod |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 2004 |
Connect to | http://dx.doi.org/10.1007/978-3-642-61943-4 |
Descript | VIII, 574 p. online resource |
I. Basic Notions of Probability Theory -- ยง 1. Axioms and Definitions -- ยง 2. Independence -- ยง 3. Conditional Probabilities and Conditional Expectations -- ยง 4. Random Functions and Random Mappings -- II. Random Sequences -- ยง 1. Preliminary Remarks -- ยง 2. Semi-Martingales and Martingales -- ยง 3. Series -- ยง 4. Markov Chains -- ยง 5. Markov Chains with a Countable Number of States -- ยง 6. Random Walks on a Lattice -- ยง 7. Local Limit Theorems for Lattice Walks -- ยง 8. Ergodic Theorems -- III. Random Functions -- ยง 1. Some Classes of Random Functions -- ยง 2. Separable Random Functions -- ยง 3. Measurable Random Functions -- ยง 4. A Criterion for the Absence of Discontinuities of the Second Kind -- ยง 5. Continuous Processes -- IV. Linear Theory of Random Processes -- ยง 1. Correlation Functions -- ยง 2. Spectral Representations of Correlation Functions -- ยง 3. A Basic Analysis of Hilbert Random Functions -- ยง 4. Stochastic Measures and Integrals -- ยง 5. Integral Representation of Random Functions -- ยง 6. Linear Transformations -- ยง 7. Physically Realizable Filters -- ยง 8. Forecasting and Filtering of Stationary Processes -- ยง 9. General Theorems on Forecasting Stationary Processes -- V. Probability Measures on Functional Spaces -- ยง 1. Measures Associated with Random Processes -- ยง 2. Measures in Metric Spaces -- ยง 3. Measures on Linear Spaces. Characteristic Functionals -- ยง 4. Measures in ?p Spaces -- ยง 5. Measures in Hilbert Spaces -- ยง 6. Gaussian Measures in a Hilbert Space -- VI. Limit Theorems for Random Processes -- ยง 1. Weak Convergences of Measures in Metric Spaces -- ยง 2. Conditions for Weak Convergence of Measures in Hilbert Spaces -- ยง 3. Sums of Independent Random Variables with Values in a Hilbert Space -- ยง 4. Limit Theorems for Continuous Random Processes -- ยง 5. Limit Theorems for Processes without Discontinuities of the Second Kind -- VII. Absolute Continuity of Measures Associated with Random Processes -- ยง 1. General Theorems on Absolute Continuity -- ยง 2. Admissible Shifts in Hilbert Spaces -- ยง 3. Absolute Continuity of Measures under Mappings of Spaces -- ยง 4. Absolute Continuity of Gaussian Measures in a Hilbert Space -- ยง 5. Equivalence and Orthogonality of Measures Associated with Stationary Gaussian Processes -- ยง 6. General Properties of Densities of Measures Associated with Markov Processes -- VIII. Measurable Functions on Hilbert Spaces -- ยง 1. Measurable Linear Functionals and Operators on Hilbert Spaces -- ยง 2. Measurable Polynomial Functions. Orthogonal Polynomials -- ยง 3. Measurable Mappings -- ยง 4. Calculation of Certain Characteristics of Transformed Measures -- Historical and Bibliographical Remarks -- Corrections