# An Introduction to Stochastic Processes and Their Applications

• Petar Todorovic
Book

Part of the Springer Series in Statistics book series (SSS)

1. Front Matter
Pages i-xiii
2. Petar Todorovic
Pages 1-33
3. Petar Todorovic
Pages 34-61
4. Petar Todorovic
Pages 62-91
5. Petar Todorovic
Pages 92-105
6. Petar Todorovic
Pages 106-128
7. Petar Todorovic
Pages 129-149
8. Petar Todorovic
Pages 150-199
9. Petar Todorovic
Pages 200-231
10. Petar Todorovic
Pages 232-257
11. Petar Todorovic
Pages 258-278
12. Back Matter
Pages 279-290

### Introduction

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.

### Keywords

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

#### Authors and affiliations

• Petar Todorovic
• 1
1. 1.Department of Statistics and Applied ProbabilityUniversity of California—Santa BarbaraSanta BarbaraUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4613-9742-7
• Copyright Information Springer-Verlag New York 1992
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4613-9744-1
• Online ISBN 978-1-4613-9742-7
• Series Print ISSN 0172-7397
• Buy this book on publisher's site
Industry Sectors
Pharma
Biotechnology