Lukacs / Birnbaum Stochastic Convergence

2. Auflage 2014
ISBN: 978-1-4832-1858-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 214 Seiten, Web PDF

Stochastic Convergence, Second Edition covers the theoretical aspects of random power series dealing with convergence problems. This edition contains eight chapters and starts with an introduction to the basic concepts of stochastic convergence. The succeeding chapters deal with infinite sequences of random variables and their convergences, as well as the consideration of certain sets of random variables as a space. These topics are followed by discussions of the infinite series of random variables, specifically the lemmas of Borel-Cantelli and the zero-one laws. Other chapters evaluate the power series whose coefficients are random variables, the stochastic integrals and derivatives, and the characteristics of the normal distribution of infinite sums of random variables. The last chapter discusses the characterization of the Wiener process and of stable processes. This book will prove useful to mathematicians and advance mathematics students.

Autoren/Hrsg.

Lukacs, Eugene

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Front Cover;1
2;Stochastic Convergence;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface to the Second Edition;8
6;Preface to the First Edition;10
7;List of Examples;12
8;Chapter 1. INTRODUCTION;14
8.1;1.1. Survey of basic concepts;14
8.2;1.2. Certain inequalities;19
8.3;1.3. Characteristic functions;26
8.4;1.4. Independence;32
8.5;1.5. Monotone classes of sets (events);36
8.6;Exercises;38
9;Chapter 2. STOCHASTIC CONVERGENCE CONCEPTS AND THEIR PROPERTIES;40
9.1;2.1. Definitions;40
9.2;2.2. Relations among the various convergence concepts;46
9.3;2.3. Convergence of sequences of mean values and of certain functions of random variables;53
9.4;2.4. Criteria for stochastic convergence;57
9.5;2.5. Further modes of stochastic convergence;63
9.6;2.6. Information convergence;67
9.7;Exercises;70
10;Chapter 3. SPACES OF RANDOM VARIABLES;73
10.1;3.1. Convergence in probability;74
10.2;3.2. Almost certain convergence;81
10.3;3.3. The spaces Lp;81
10.4;3.4. The space of distribution functions;83
10.5;Exercises;87
11;Chapter 4. INFINITE SERIES OF RANDOM VARIABLES AND RELATED TOPICS;89
11.1;4.1. The lemmas of Borel-Cantelli and the zero-one laws;89
11.2;4.2. Convergence of series;93
11.3;4.3. Some limit theorems;107
11.4;Exercises;123
12;Chapter 5. RANDOM POWER SERIES;125
12.1;5.1. Definition and convergence of random power series;125
12.2;5.2. The radius of convergence of a random power series;130
12.3;5.3. Random power series with identically distributed coefficients;135
12.4;5.4. Random power series with independent coefficients;140
12.5;5.5. The analytic continuation of random power series;143
12.6;5.6. Random entire functions;148
12.7;Exercises;153
13;Chapter 6. STOCHASTIC INTEGRALS AND DERIVATIVES;156
13.1;6.1. Some definitions concerning stochastic processes;156
13.2;6.2. Definition and existence of stochastic integrals;158
13.3;6.3. L2-continuity and differentiation of stochastic processes;165
13.4;Exercises;167
14;Chapter 7. CHARACTERIZATION OF THE NORMAL DISTRIBUTION BY PROPERTIES OF INFINITE SUMS OF RANDOM VARIABLES;170
14.1;7.1. Identically distributed linear forms;170
14.2;7.2. A linear form and a monomial having the same distribution;174
14.3;7.3. Independently distributed infinite sums;181
14.4;Exercises;183
15;Chapter 8. CHARACTERIZATION OF SOME STOCHASTIC PROCESSES;185
15.1;8.1. Independence and a regression property of two stochastic integrals;185
15.2;8.2. Identically distributed stochastic integrals;188
15.3;8.3. Identity of the distribution of a stochastic integral and the increment of a process;191
15.4;8.4. Characterization of stable processes;195
15.5;Exercises;202
16;References;204
17;Index;208