# A Graduate Course in Probability

## 1st Edition

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eBook ISBN: 9781483220505
Published Date: 1st January 1967
Page Count: 288
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## Description

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.

Preface

Chapter 1 Probability Spaces

1.1 Sigma Fields

1.2 Probability Measures

1.3 Random Variables

Chapter 2 Probability Distributions

2.1 Univariate Distribution Functions

2.2 Multivariate Distribution Functions

2.3 Distribution of a Set of Infinitely Many Random Variables

2.4 Expectation

2.5 Characteristic Functions

Chapter 3 Stochastic Independence

3.1 Independent Events

3.2 Independent Random Variables

3.3 The Zero-One Law

Chapter 4 Basic Limiting Operations

4.1 Convergence of Distribution Functions

4.2 The Continuity Theorem

4.3 Refinements of the Continuity Theorem for Nonvanishing Characteristic Functions

4.4 The Four Types of Convergence: Almost Sure, in Law, in Probability, and in rth Mean

Chapter 5 Strong Limit Theorems for Independent Random Variables

5.1 Almost Sure Convergence of Series of Independent Random Variables

5.2 Proof that Convergence in Law of a Series of Independent Random Variables Implies Almost Sure Convergence

5.3 The Strong Law of Large Numbers

5.4 The Glivenko-Cantelli Theorem

5.5 Inequalities for the Law of the Iterated Logarithm

5.6 The Law of the Iterated Logarithm

Chapter 6 The Central Limit Theorem

6.1 Infinitely Divisible Distributions

6.2 Canonical Representation of Infinitely Divisible Characteristic Functions

6.3 Convergence of Infinitely Divisible Distribution Functions

6.4 Infinitesimal Systems of Random Variables

6.5 The General Limit Theorem for Sequences of Sums of Independent Random Variables

6.6 Convergence to the Normal and Poisson Distributions

Chapter 7 Conditional Expectation and Martingale Theory

7.1 Conditional Expectation

7.2 Martingales and Submartingales

7.3 Martingale and Submartingale Convergence Theorems

7.4 Brownian Motion

Chapter 8 An Introduction to Stochastic Processes and, in Particular, Brownian Motion

8.1 Probability Measures over Function Spaces

8.2 Separable Stochastic Processes

8.3 Continuity and NonrectifiabiUty of Almost All Sample Functions of Separable Brownian Motion

8.4 The Law of the Iterated Logarithm for Separable Brownian Motion

Index

No. of pages:
288
Language:
English
Published:
Imprint:
eBook ISBN:
9781483220505