An Introduction to Stochastic Processes and Their Applications

1. Front Matter

   Pages i-xiii

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2. Basic Concepts and Definitions

   • Petar Todorovic

   Pages 1-33

3. The Poisson Process and Its Ramifications

   • Petar Todorovic

   Pages 34-61

4. Elements of Brownian Motion

   • Petar Todorovic

   Pages 62-91

5. Gaussian Processes

   • Petar Todorovic

   Pages 92-105

6. L ₂ Space

   • Petar Todorovic

   Pages 106-128

7. Second-Order Processes

   • Petar Todorovic

   Pages 129-149

8. Spectral Analysis of Stationary Processes

   • Petar Todorovic

   Pages 150-199

9. Markov Processes I

   • Petar Todorovic

   Pages 200-231

10. Markov Processes II: Application of Semigroup Theory

    • Petar Todorovic

    Pages 232-257

11. Discrete Parameter Martingales

    • Petar Todorovic

    Pages 258-278

12. Back Matter

    Pages 279-290

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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.

• Brownian motion
• Gaussian process
• Markov chain
• Markov process
• Markov property
• Martingale
• Normal distribution
• Ornstein-Uhlenbeck process
• Poisson process
• Stochastic proc
• hitting time
• law of the iterated logarithm
• point process
• stochastic process

• Department of Statistics and Applied Probability, University of California—Santa Barbara, Santa Barbara, USA

  Petar Todorovic

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