Author | Todorovic, Petar. author |
---|---|
Title | An Introduction to Stochastic Processes and Their Applications [electronic resource] / by Petar Todorovic |
Imprint | New York, NY : Springer New York, 1992 |
Connect to | http://dx.doi.org/10.1007/978-1-4613-9742-7 |
Descript | XIV, 289 p. online resource |
1 Basic Concepts and Definitions -- 1.1. Definition of a Stochastic Process -- 1.2. Sample Functions -- 1.3. Equivalent Stochastic Processes -- 1.4. Kolmogorov Construction -- 1.5. Principal Classes of Random Processes -- 1.6. Some Applications -- 1.7. Separability -- 1.8. Some Examples -- 1.9. Continuity Concepts -- 1.10. More on Separability and Continuity -- 1.11. Measurable Random Processes -- Problems and Complements -- 2 The Poisson Process and Its Ramifications -- 2.1. Introduction -- 2.2. Simple Point Process on R+ -- 2.3. Some Auxiliary Results -- 2.4. Definition of a Poisson Process -- 2.5. Arrival Times ?k -- 2.6. Markov Property of N(t) and Its Implications -- 2.7. Doubly Stochastic Poisson Process -- 2.8. Thinning of a Point Process -- 2.9. Marked Point Processes -- 2.10. Modeling of Floods -- Problems and Complements -- 3 Elements of Brownian Motion -- 3.1. Definitions and Preliminaries -- 3.2. Hitting Times -- 3.3. Extremes of ?(t) -- 3.4. Some Properties of the Brownian Paths -- 3.5. Law of the Iterated Logarithm -- 3.6. Some Extensions -- 3.7. The Ornstein-Uhlenbeck Process -- 3.8. Stochastic Integration -- Problems and Complements -- 4 Gaussian Processes -- 4.1. Review of Elements of Matrix Analysis -- 4.2. Gaussian Systems -- 4.3. Some Characterizations of the Normal Distribution -- 4.4. The Gaussian Process -- 4.5. Markov Gaussian Process -- 4.6. Stationary Gaussian Process -- Problems and Complements -- 5 L2 Space -- 5.1. Definitions and Preliminaries -- 5.2. Convergence in Quadratic Mean -- 5.3. Remarks on the Structure of L2 -- 5.4. Orthogonal Projection -- 5.5. Orthogonal Basis -- 5.6. Existence of a Complete Orthonormal Sequence in L2 -- 5.7. Linear Operators in a Hilbert Space -- 5.8. Projection Operators -- Problems and Complements -- 6 Second-Order Processes -- 6.1. Covariance Function C(s,t) -- 6.2. Quadratic Mean Continuity and Differentiability -- 6.3. Eigenvalues and Eigenfunctions of C(s, t) -- 6.4. Karhunen-Loeve Expansion -- 6.5. Stationary Stochastic Processes -- 6.6. Remarks on the Ergodicity Property -- Problems and Complements -- 7 Spectral Analysis of Stationary Processes -- 7.1. Preliminaries -- 7.2. Proof of the Bochner-Khinchin and Herglotz Theorems -- 7.3. Random Measures -- 7.4. Process with Orthogonal Increments -- 7.5. Spectral Representation -- 7.6. Ramifications of Spectral Representation -- 7.7. Estimation, Prediction, and Filtering -- 7.8. An Application -- 7.9. Linear Transformations -- 7.10. Linear Prediction, General Remarks -- 7.11. The Wold Decomposition -- 7.12. Discrete Parameter Processes -- 7.13. Linear Prediction -- 7.14. Evaluation of the Spectral Characteristic ?(?, h) -- 7.15. General Form of Rational Spectral Density -- Problems and Complements -- 8 Markov Processes I -- 8.1. Introduction -- 8.2. Invariant Measures -- 8.3. Countable State Space -- 8.4. Birth and Death Process -- 8.5. Sample Function Properties -- 8.6. Strong Markov Processes -- 8.7. Structure of a Markov Chain -- 8.8. Homogeneous Diffusion -- Problems and Complements -- 9 Markov Processes II: Application of Semigroup Theory -- 9.1. Introduction and Preliminaries -- 9.2. Generator of a Semigroup -- 9.3. The Resolvent -- 9.4. Uniqueness Theorem -- 9.5. The Hille-Yosida Theorem -- 9.6. Examples -- 9.7. Some Refinements and Extensions -- Problems and Complements -- 10 Discrete Parameter Martingales -- 10.1. Conditional Expectation -- 10.2. Discrete Parameter Martingales -- 10.3. Examples -- 10.4. The Upcrossing Inequality -- 10.5. Convergence of Submartingales -- 10.6. Uniformly Integrable Martingales -- Problems and Complements