Probability: A Graduate Course

Preface

Contents

Outline of Contents

Notation and Symbols

1 Introductory Measure Theory

1 Probability Theory: An Introduction

2 Basics from Measure Theory

3 The Probability Space

4 Independence; Conditional Probabilities

5 The Kolmogorov Zero-one Law

6 Problems

2 Random Variables

1 Definition and Basic Properties

2 Distributions

3 Random Vectors; Random Elements

4 Expectation; Definitions and Basics

5 Expectation; Convergence

6 Indefinite Expectations

7 A Change of Variables Formula

8 Moments, Mean, Variance

9 Product Spaces; Fubini’s Theorem

10 Independence

11 The Cantor Distribution

12 Tail Probabilities and Moments

13 Conditional Distributions

14 Distributions with Random Parameters

15 Sums of a Random Number of Random Variables

16 Random Walks; Renewal Theory

17 Extremes; Records

18 Borel-Cantelli Lemmas

19 A Convolution Table

20 Problems

3 Inequalities

1 Tail Probabilities Estimated via Moments

2 Moment Inequalities

3 Covariance; Correlation

4 Interlude on Lp-spaces

5 Convexity

6 Symmetrization

7 Probability Inequalities for Maxima

8 The Marcinkiewics-Zygmund Inequalities

9 Rosenthal’s Inequality

10 Problems

4 Characteristic Functions

1 Definition and Basics

2 Some Special Examples

3 Two Surprises

4 Refinements

5 Characteristic Functions of Random Vectors

6 The Cumulant Generating Function

7 The Probability Generating Function

8 The Moment Generating Function

9 Sums of a Random Number of Random Variables

10 The Moment Problem

11 Problems

5 Convergence

1 Definitions

2 Uniqueness

3 Relations Between Convergence Concepts

4 Uniform Integrability

5 Convergence of Moments

6 Distributional Convergence Revisited

7 A Subsequence Principle

8 Vague Convergence; Helly’s Theorem

9 Continuity Theorems

10 Convergence of Functions of Random Variables

11 Convergence of Sums of Sequences

12 Cauchy Convergence

13 Skorohod’s Representation Theorem

14 Problems

6 The Law of Large Numbers

1 Preliminaries

2 A Weak Law for Partial Maxima

3 The Weak Law of Large Numbers

4 A Weak Law Without Finite Mean

5 Convergence of Series

6 The Strong Law of Large Numbers

7 The Marcinkiewicz-Zygmund Strong Law

8 Randomly Indexed Sequences

9 Applications

10 Uniform Integrability; Moment Convergence

11 Complete Convergence

12 Some Additional Results and Remarks

13 Problems

7 The Central Limit Theorem

1 The i.i.d. Case

2 The Lindeberg-Levy-Feller Theorem

3 Anscombe’s Theorem

4 Applications

5 Uniform Integrability; Moment Convergence

6 Remainder Term Estimates

7 Some Additional Results and Remarks

8 Problems

8 The Law of the Iterated Logarithm

1 The Kolmogorov and Hartman-Wintner LILs

2 Exponential Bounds

3 Proof of the Hartman-Wintner Theorem

4 Proof of the Converse

5 The LIL for Subsequences

6 Cluster Sets

7 Some Additional Results and Remarks

8 Problems

9 Limit Theorems; Extensions and Generalizations

1 Stable Distributions

2 The Convergence to Types Theorem

3 Domains of Attraction

4 Infinitely Divisible Distributions

5 Sums of Dependent Random Variables

6 Convergence of Extremes

7 The Stein-Chen Method

8 Problems

10 Martingales

1 Conditional Expectation

2 Martingale Definitions

3 Examples

4 Orthogonality

5 Decompositions

6 Stopping Times

7 Doob’s Optional Sampling Theorem

8 Joining and Stopping Martingales

9 Martingale Inequalities

10 Convergence

11 The Martingale { E( Z | Fn)}

12 Regular Martingales and Submartingales

13 The Kolmogorov Zero-one Law

14 Stopped Random Walks

15 Regularity

16 Reversed Martingales and Submartingales

17 Problems

A Some Useful Mathematics

1 Taylor Expansion

2 Mill’s Ratio

3 Sums and Integrals

4 Sums and Products

5 Convexity; Clarkson’s Inequality

6 Convergence of (Weighted) Averages

7 Regularly and Slowly Varying Functions

8 Cauchy’s Functional Equation

9 Functions and Dense Sets

References

Index