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girardin valérie; limnios nikolaos - applied probability

Applied Probability From Random Sequences to Stochastic Processes

Name: Applied Probability
Price: 77,88 € EUR
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Author: Girardin Valérie; Limnios Nikolaos
ISBN: 9783319974118

girardin valérie; limnios nikolaos

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Dettagli

Genere:Libro

Lingua: Inglese

Editore:

Springer

Pubblicazione: 09/2018

Edizione: 1st ed. 2018

Trama

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.

Sommario

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

Autore

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.