Preface

1 Measure Theory 1

1.1 Probability Spaces 1

1.2 Distributions 9

1.3 Random Variables 14

1.4 Integration 17

1.5 Properties of the Integral 23

1.6 Expected Value 27

1.6.1 Inequalities 27

1.6.2 Integration to the Limit 29

1.6.3 Computing Expected Values 30

1.7 Product Measures, Fubini's Theorem 36

2 Laws of Large Numbers 41

2.1 Independence 41

2.1.1 Sufficient Conditions for Independence 43

2.1.2 Independence, Distribution, and Expectation 45

2.1.3 Sums of Independent Random Variables 47

2.1.4 Constructing Independent Random Variables 50

2.2 Weak Laws of Large Numbers 53

2.2.1 L2 Weak Laws 53

2.2.2 Triangular Arrays 56

2.2.3 Truncation 59

2.3 Borel-Cantelli Lemmas 64

2.4 Strong Law of Large Numbers 73

2.5 Convergence of Random Series 78

2.5.1 Rates of Convergence 82

2.5.2 Infinite Mean 84

2.6 Large Deviations 86

3 Central Limit Theorems 94

3.1 The De Moivre-Laplace Theorem 94

3.2 Weak Convergence 97

3.2.1 Examples 97

3.2.2 Theory 100

3.3 Characteristic Functions 106

3.3.1 Definition, Inversion Formula 106

3.3.2 Weak Convergence 112

3.3.3 Moments and Derivatives 114

3.3.4 Polya's Criterion 118

3.3.5 The Moment Problem 120

3.4 Central Limit Theorems 124

3.4.1 i.i.d. Sequences 124

3.4.2 Triangular Arrays 129

3.4.3 Prime Divisors (Erdos-Kac) 133

3.4.4 Rates of Convergence (Berry-Esseen) 137

3.5 Local Limit Theorems 141

3.6 Poisson Convergence 146

3.6.1 The Basic Limit Theorem 146

3.6.2 Two Examples with Dependence 151

3.6.3 Poisson Processes 154

3.7 Stable Laws 158

3.8 Infinitely Divisible Distributions 169

3.9 Limit Theorems in Rd 172

4 Random Walks 179

4.1 Stopping Times 179

4.2 Recurrence 189

4.3 Visits to 0, Arcsine Laws 201

4.4 Renewal Theory 208

5 Martingales 221

5.1 Conditional Expectation 221

5.1.1 Examples 223

5.1.2 Properties 226

5.1.3 Regular Conditional Probabilities 230

5.2 Martingales, Almost Sure Convergence 232

5.3 Examples 239

5.3.1 Bounded Increments 239

5.3.2 Polya's Urn Scheme 241

5.3.3 Radon-Nikodym Derivatives 242

5.3.4 Branching Processes 245

5.4 Doob's Inequality, Convergence in Lp 249

5.4.1 Square Integrable Martingales 254

5.5 Uniform Integrability, Convergence in L1 258

5.6 Backwards Martingales 264

5.7 Optional Stopping Theorems 269

6 Markov Chains 274

6.1 Definitions 274

6.2 Examples 277

6.3 Extensions of the Markov Property 282

6.4 Recurrence and Transience 288

6.5 Stationary Measures 296

6.6 Asymptotic Behavior 307

6.7 Periodicity, Tail σ -field 314

6.8 General State Space 318

6.8.1 Recurrence and Transience 322

6.8.2 Stationary Measures 323

6.8.3 Convergence Theorem 324

6.8.4 GI/G/1 Queue 325

7 Ergodic Theorems 328

7.1 Definitions and Examples 328

7.2 Birkhoff's Ergodic Theorem 333

7.3 Recurrence 338

7.4 A Subadditive Ergodic Theorem 342

7.5 Applications 347

8 Brownian Motion 353

8.1 Definition and Construction 353

8.2 Markov Property, Blumenthal's 0-1 Law 359

8.3 Stopping Times, Strong Markov Property 365

8.4 Path Properties 370

8.4.1 Zeros of Brownian Motion 370

8.4.2 Hitting Times 371

8.4.3 Levy's Modulus of Continuity 375

8.5 Martingales 376

8.5.1 Multidimensional Brownian Motion 380

8.6 Donsker's Theorem 382

8.7 Empirical Distributions, Brownian Bridge 391

8.8 Laws of the Iterated Logarithm 396

Appendix A Measure Theory Details 401

A.1 Caratheodory's Extension Theorem 401

A.2 Which Sets Are Measurable? 407

A.3 Kolmogorov's Extension Theorem 410

A.4 Radon-Nikodym Theorem 412

A.5 Differentiating under the Integral 416

References 419

Index 425