Applied Probability: From Random Sequences to Stochastic Processes

ISBN-13: 9783319974118 / Angielski / Twarda / 2018 / 260 str.

Valerie Girardin; Nikolaos Limnios

Recenzje

"The primary audience of the book are students studying mathematics at a university and Ph.D. students in applied mathematics. It can also be served as a course at master's and doctoral's levels in applied probability. And finally, it is conceived as a support for the researchers and engineers dealing with stochastic modelling." (Anatoliy Swishchuk, zbMATH 1434.60002, 2020)
"It is intended for students (at master or PhD level) in applied mathematics as well as researchers and engineers dealing with stochastic modeling issues. ... Finally, the readers who want to get their hands on the different notions presented in the book will find at the end of each section a series of exercises with their solutions." (Julien Poisat, Mathematical Reviews, October, 2019)

Spis treści

Notation iii

Preface ix

1 Independent Random Sequences 1

1.1 Denumerable Sequences . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Sequences of Events . . . . . . . . . . . . . . . . . . . . . 7

1.1.2 Independence . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Analytic Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.1 Generating Functions . . . . . . . . . . . . . . . . . . . . 12

1.2.2 Characteristic Functions . . . . . . . . . . . . . . . . . . . 13

1.2.3 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . 15

1.2.4 Moment Generating Functions and Cram´er Transforms . 17

1.2.5 From Entropy to Entropy Rate . . . . . . . . . . . . . . . 19

1.3 Sums and Random Sums . . . . . . . . . . . . . . . . . . . . . . . 23

1.3.1 Sums of Independent Variables . . . . . . . . . . . . . . . 23

1.3.2 Random Sums . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3.3 Random Walks . . . . . . . . . . . . . . . . . . . . . . . . 29

1.4 Convergence of Random Sequences . . . . . . . . . . . . . . . . . 30

1.4.1 Different Types of Convergence . . . . . . . . . . . . . . . 30

1.4.2 Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . 33

1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2 Conditioning and Martingales 51

2.1 Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.1.1 Conditioning with Respect to an Event . . . . . . . . . . 52

2.1.2 Conditional Probabilities . . . . . . . . . . . . . . . . . . 53

2.1.3 Conditional Distributions . . . . . . . . . . . . . . . . . . 56

2.1.4 Conditional Expectation . . . . . . . . . . . . . . . . . . . 57

2.1.5 Conditioning and Independence . . . . . . . . . . . . . . . 63

2.1.6 Practical Determination . . . . . . . . . . . . . . . . . . . 65

2.2 The Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.3 Stopping Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.4 Discrete-Time Martingales . . . . . . . . . . . . . . . . . . . . . . 73

2.4.1 Definitions and Properties . . . . . . . . . . . . . . . . . . 73

2.4.2 Classical Inequalities . . . . . . . . . . . . . . . . . . . . . 78

2.4.3 Martingales and Stopping Times . . . . . . . . . . . . . . 81

2.4.4 Convergence of Martingales . . . . . . . . . . . . . . . . . 84

2.4.5 Square Integrable Martingales . . . . . . . . . . . . . . . . 86

2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3 Markov Chains 99

3.1 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.1.1 Transition Functions with Examples . . . . . . . . . . . . 99

3.1.2 Martingales and Markov Chains . . . . . . . . . . . . . . 107

3.1.3 Stopping Times and Markov Chains . . . . . . . . . . . . 109

3.2 Classification of States . . . . . . . . . . . . . . . . . . . . . . . . 111

3.3 Stationary Distribution and Asymptotic Behavior . . . . . . . . . 116

3.4 Periodic Markov chains . . . . . . . . . . . . . . . . . . . . . . . 123

3.5 Finite Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . 127

3.5.1 Specific Properties . . . . . . . . . . . . . . . . . . . . . . 127

3.5.2 Application to Reliability . . . . . . . . . . . . . . . . . . 132

3.6 Branching Processes . . . . . . . . . . . . . . . . . . . . . . . . . 135

3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

4 Continuous Time Stochastic Processes 153

4.1 General Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

4.2 Stationarity and Ergodicity . . . . . . . . . . . . . . . . . . . . . 160

4.3 Processes with Independent Increments . . . . . . . . . . . . . . 166

4.4 Point Processes on the Line . . . . . . . . . . . . . . . . . . . . . 168

4.4.1 Basics on General Point Processes . . . . . . . . . . . . . 169

4.4.2 Renewal Processes . . . . . . . . . . . . . . . . . . . . . . 171

4.4.3 Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . 175

4.4.4 Asymptotic Results for Renewal Processes . . . . . . . . . 177

4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

5 Markov and Semi-Markov Processes 189

5.1 Jump Markov Processes . . . . . . . . . . . . . . . . . . . . . . . 189

5.1.1 Markov Processes . . . . . . . . . . . . . . . . . . . . . . . 189

5.1.2 Transition Functions . . . . . . . . . . . . . . . . . . . . . 192

5.1.3 Infinitesimal Generators and Kolmogorov’s Equations . . 195

5.1.4 Embedded Chains and Classification of States . . . . . . . 197

5.1.5 Stationary Distribution and Asymptotic Behavior . . . . 203

5.2 Semi-Markov Processes . . . . . . . . . . . . . . . . . . . . . . . 207

5.2.1 Markov Renewal Processes . . . . . . . . . . . . . . . . . 207

5.2.2 Classification of States and Asymptotic Behavior . . . . . 210

5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Further Reading 225

O autorze

Valérie Girardin received her Ph.D. in Probability from the Université Paris-Sud in Orsay, France. She teaches analysis, probability and statistics to various levels of students, including future secondary school teachers in mathematics, future engineers and researchers. Her research interests include diverse aspects of stochastic processes, from theory to applied statistics, with a particular interest in information theory and biology.

Nikolaos Limnios graduated from the Aristotle University of Thessaloniki and Polytechnic School of Thesaloniki, Greece. He received his Ph.D. and his Doctorat d’Etat  from the Université de Technologie de Compiègne (UTC), France, where he is now a full professor. He teaches probability,  statistics and stochastic processes to future engineers. His research interests in stochastic processes  and statistics include Markov, semi-Markov processes, branching processes, random evolutions and their applications in biology, reliability, earthquake, population evolutions, among other topics. 

 

Blurb

This textbook addresses postgraduate students in applied mathematics, probability, and statistics, as well as computer scientists, biologists, physicists and economists, who are seeking a rigorous introduction to applied stochastic processes. Pursuing a pedagogic approach, the content follows a path of increasing complexity, from the simplest random sequences to the advanced stochastic processes. Illustrations are provided from many applied fields, together with connections to ergodic theory, information theory, reliability and insurance. The main content is also complemented by a wealth of examples and exercises with solutions.