1 Measure Theory

1.1 Probability Spaces

Measures on Rd

Exercises

1.2 Distributions

Exercises

1.3 Random Variables

Exercises

1.4 Integration

Exercises

1.5 Properties of the Integral

Exercises

1.6 Expected Value

1.6.1 Inequalities

1.6.2 Integration to the Limit

1.6.3 Computing Expected Values

Exercises

1.7 Product Measures, Fubini's Theorem

Exercises

2 Laws of Large Numbers

2.1 Independence

2.1.1 Sufficient Conditions for Independence

2.1.2 Independence, Distribution, and Expectation

2.1.3 Sums of Independent Random Variables

2.1.4 Constructing Independent Random Variables

Exercises

2.2 Weak Laws of Large Numbers

2.2.1 L2 Weak Laws

2.2.2 Triangular Arrays

2.2.3 Truncation

Exercises

2.3 Borel-Cantelli Lemmas

Exercises

2.4 Strong Law of Large Numbers

Exercises

2.5 Convergence of Random Series

2.5.1 Rates of Convergence

2.5.2 Infinite Mean

Exercises

2.6 Large Deviations

3 Central Limit Theorems

3.1 The De Moivre-Laplace Theorem

Exercises

3.2 Weak Convergence

3.2.1 Examples

3.2.2 Theory

Exercises

3.3 Characteristic Functions

3.3.1 Definition, Inversion Formula

3.3.2 Weak Convergence

3.3.3 Moments and Derivatives

3.3.4 Polya's Criterion

3.3.5 The Moment Problem

3.4 Central Limit Theorems

3.4.1 i.i.d. Sequences

Exercises

3.4.2 Triangular Arrays

Exercises

3.4.3 Prime Divisors (Erdos-Kac)

3.4.4 Rates of Convergence (Berry-Esseen)

3.5 Local Limit Theorems

3.6 Poisson Convergence

3.6.1 The Basic Limit Theorem

3.6.2 Two Examples with Dependence

3.6.3 Poisson Processes

3.7 Stable Laws

Exercises

3.8 Infinitely Divisible Distributions

3.9 Limit Theorems in Rd

4 Random Walks

4.1 Stopping Times

4.2 Recurrence

4.3 Visits to 0, Arcsine Laws

4.4 Renewal Theory

5 Martingales

5.1 Conditional Expectation

5.1.1 Examples

5.1.2 Properties

5.1.3 Regular Conditional Probabilities

5.2 Martingales, Almost Sure Convergence

Exercises

5.3 Examples

5.3.1 Bounded Increments

5.3.2 Polya's Urn Scheme

5.3.3 Radon-Nikodym Derivatives

5.3.4 Branching Processes

5.4 Doob's Inequality, Convergence in Lp

5.4.1 Square Integrable Martingales

5.5 Uniform Integrability, Convergence in L1

5.6 Backwards Martingales

Exercises

5.7 Optional Stopping Theorems

Exercises

6 Markov Chains

6.1 Definitions

6.2 Examples

Exercises

6.3 Extensions of the Markov Property

Exercises

6.4 Recurrence and Transience

6.5 Stationary Measures

6.6 Asymptotic Behavior

Exercises

6.7 Periodicity, Tail sigma -field

6.8 General State Space

6.8.1 Recurrence and Transience

6.8.2 Stationary Measures

6.8.3 Convergence Theorem

6.8.4 GI/G/1 Queue

7 Ergodic Theorems

7.1 Definitions and Examples

Exercises

7.2 Birkhoff's Ergodic Theorem

7.3 Recurrence

7.4 A Subadditive Ergodic Theorem

7.5 Applications

8 Brownian Motion

8.1 Definition and Construction

8.2 Markov Property, Blumenthal's 0-1 Law

8.3 Stopping Times, Strong Markov Property

8.4 Path Properties

8.4.1 Zeros of Brownian Motion

8.4.2 Hitting Times

8.4.3 Levy's Modulus of Continuity

8.5 Martingales

8.5.1 Multidimensional Brownian Motion

8.6 Donsker's Theorem

8.7 Empirical Distributions, Brownian Bridge

8.8 Laws of the Iterated Logarithm

Appendix A: Measure Theory Details

A.1 Caratheodory's Extension Theorem

A.2 Which Sets Are Measurable?

A nonmeasurable subset of [0,1)

Banach-Tarski theorem

Solovay’s theorem

A.3 Kolmogorov's Extension Theorem

A.4 Radon-Nikodym Theorem

A.5 Differentiating under the Integral

References

Index