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.

This text on stochastic processes and their applications is based on a set of lectures given during the past several years at the University of California, Santa Barbara (UCSB). It is an introductory graduate course designed for classroom purposes. Its objective is to provide graduate students of statistics with an overview of some basic methods and techniques in the theory of stochastic processes. The only prerequisites are some rudiments of measure and integration theory and an intermediate course in probability theory. There are more than 50 examples and applications and 243 problems and complements which appear at the end of each chapter. The book consists of 10 chapters. Basic concepts and definitions are pro­ vided in Chapter 1. This chapter also contains a number of motivating ex­ amples and applications illustrating the practical use of the concepts. The last five sections are devoted to topics such as separability, continuity, and measurability of random processes, which are discussed in some detail. The concept of a simple point process on R+ is introduced in Chapter 2. Using the coupling inequality and Le Cam's lemma, it is shown that if its counting function is stochastically continuous and has independent increments, the point process is Poisson. When the counting function is Markovian, the sequence of arrival times is also a Markov process. Some related topics such as independent thinning and marked point processes are also discussed. In the final section, an application of these results to flood modeling is presented.