Tucker / Birnbaum / Lukacs A Graduate Course in Probability

1. Auflage 2014
ISBN: 978-1-4832-2050-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 288 Seiten, Web PDF

Probability and Mathematical Statistics: A Series of Monographs and Textbooks: A Graduate Course in Probability presents some of the basic theorems of analytic probability theory in a cohesive manner. This book discusses the probability spaces and distributions, stochastic independence, basic limiting operations, and strong limit theorems for independent random variables. The central limit theorem, conditional expectation and martingale theory, and Brownian motion are also elaborated. The prerequisite for this text is knowledge of real analysis or measure theory, particularly the Lebesgue dominated convergence theorem, Fubini's theorem, Radon-Nikodym theorem, Egorov's theorem, monotone convergence theorem, and theorem on unique extension of a sigma-finite measure from an algebra to the sigma-algebra generated by it. This publication is suitable for a one-year graduate course in probability given in a mathematics program and preferably for students in their second year of graduate work.

Autoren/Hrsg.

Tucker, Howard G.

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Front Cover;1
2;A Graduate Course in Probability;4
3;Copyright Page;5
4;Table of Contents;12
5;Dedication;6
6;Preface;8
7;CHAPTER 1. Probability Spaces;16
7.1;1.1 Sigma Fields;16
7.2;1.2 Probability Measures;20
7.3;1.3 Random Variables;24
8;CHAPTER 2.
Probability Distributions;30
8.1;2.1. Univariate Distribution Functions;30
8.2;2.2. Multivariate Distribution Functions;38
8.3;2.3. Distribution of a Set of Infinitely Many Random Variables;44
8.4;2.4. Expectation;49
8.5;2.5. Characteristic Functions;56
9;CHAPTER 3.
Stochastic Independence;72
9.1;3.1. Independent Events;72
9.2;3.2. Independent Random Variables;76
9.3;3.3. The Zero-One Law;85
10;CHAPTER 4.
Basic Limiting Operations;92
10.1;4.1. Convergence of Distribution Functions;92
10.2;4.2. The Continuity Theorem;103
10.3;4.3. Refinements of the Continuity Theorem for Nonvanishing Characteristic Functions;107
10.4;4.4. The Four Types of Convergence: Almost Sure, in Law, in Probability, and in rth Mean;114
11;CHAPTER 5.
Strong Limit Theorems for Independent Random Variables;122
11.1;5.1. Almost Sure Convergence of Series of Independent Random Variables;122
11.2;5.2. Proof that Convergence in Law of a Series of Independent Random Variables Implies Almost Sure Convergence;130
11.3;5.3. The Strong Law of Large Numbers;137
11.4;5.4. The Glivenko-Cantelli Theorem;141
11.5;5.5. Inequalities for the Law of the Iterated Logarithm;144
11.6;5.6. The Law of the Iterated Logarithm;152
12;CHAPTER 6.
The Central Limit Theorem;162
12.1;6.1. Infinitely Divisible Distributions;162
12.2;6.2. Canonical Representation of Infinitely Divisible Characteristic Functions;168
12.3;6.3 Convergence of Infinitely Divisible Distribution Functions;176
12.4;6.4. Infinitesimal Systems of Random Variables;182
12.5;6.5. The General Limit Theorem for Sequences of Sums of Independent Random Variables;192
12.6;6.6. Convergence to the Normal and Poisson Distributions;209
13;CHAPTER 7.
Conditional Expectation and Martingale Theory;224
13.1;7.1. Conditional Expectation;224
13.2;7.2. Martingales and Submartingales;235
13.3;7.3. Martingale and Submartingale Convergence Theorems;244
13.4;7.4. Brownian Motion;254
14;CHAPTER 8. An Introduction to Stochastic Processes and, in Particular, Brownian Motion;260
14.1;8.1 Probability Measures over Function Spaces;260
14.2;8.2 Separable Stochastic Processes;264
14.3;8.3 Continuity and Nonrectifiability of Almost All Sample Functions of Separable Brownian Motion;269
14.4;8.4. The Law of the Iterated Logarithm for Separable Brownian Motion;280
15;Suggested Reading;285
16;Index;286