Statistics Topics Index: Probability: Theory and Examples, 3Ed

Probability: Theory and Examples, 3Ed

Courses that use this book:

[Fall 2006] Statistics 205A -- Probability Theory

Probability: Theory And Examples

Probability: Theory And Examples, 3ed.
Author(s): Richard Durrett.
ISBN: 0534424414.
Year: 2005.
Publisher: Thomson Brooks/Cole.

1 Laws Of Large Numbers 1

1.1 Basic Definitions 1

1.2 Random Variables 8

1.3 Expected Value 12

1.3.1 Inequalities For Expected Value 12

1.3.2 Integration To The Limit 15

1.3.3 Computing Expected Values 17

1.4 Independence 22

1.4.1 Sufficient Conditions For Independence 23

1.4.2 Independence, Distribution, And Expectation 26

1.4.3 Constructing Independent Random Variables 31

1.5 Weak Laws Of Large Numbers 34

1.5.1 L2 Weak Laws 34

1.5.2 Triangular Arrays 37

1.5.3 Truncation Of Random Variables 40

1.6 Borel-Cantelli Lemmas 46

1.7 Strong Law Of Large Numbers 55

1.8 Convergence Of Random Series 60

1.9 Large Deviations For The Strong Law Of Large Numbers 69

2 Central Limit Theorems 77

2.1 The De Moivre-Laplace Theorem 77

2.2 Weak Convergence 80

2.2.1 Examples 80

2.2.2 Theory Of Weak Convergence 83

2.3 Characteristic Functions 89

2.3.a Definition, Inversion Formula For Characteristic Function 90

2.3.b Weak Convergence And Characteristic Functions 97

2.3.c Moments And Derivatives And Characteristic Functions 99

2.3.d Polya's Criterion 102

2.3.e The Moment Problem 105

2.4 Central Limit Theorems 110

2.4.a I.I.D. Sequences 110

2.4.b Triangular Arrays 114

2.4.c Prime Divisors (Erdös-Kac) 119

2.4.d Rates Of Convergence (Berry-Esseen) 124

2.5 Local Limit Theorems 129

2.6 Poisson Convergence 135

2.6.a Basic Limit Theorem For Poisson Convergence 135

2.6.b Two Examples With Dependence 140

2.6.c Poisson Processes 143

2.7 Stable Laws 147

2.8 Infinitely Divisible Distributions 159

2.9 Limit Theorems In R^D 162

3 Random Walks 171

3.1 Stopping Times 171

3.2 Recurrence 182

3.3 Visits To 0, Arcsine Laws 195

3.4 Renewal Theory 202

4 Martingales 217

4.1 Conditional Expectation 217

4.1.1 Examples 219

4.1.2 Properties Of Conditional Expectation 222

4.1.3 Regular Conditional Probabilities 227

4.2 Martingales And Almost Sure Convergence 228

4.3 Examples 236

4.3.1 Martingales With Bounded Increments 236

4.3.2 Polya's Urn Scheme 238

4.3.3 Radon-Nikodym Derivatives 239

4.3.4 Branching Processes As Martingales 243

4.4 Doob's Inequality, Lp Convergence 246

4.4.1 Square Integrable Martingales 252

4.5 Uniform Integrability, Convergence In L1 256

4.6 Backwards Martingales 262

5 Markov Chains 274

5.1 Definitions And Examples 274

5.2 Extensions Of The Markov Property 282

5.3 Recurrence And Transience 288

5.4 Stationary Measures 296

5.5 Asymptotic Behavior Of Markov Chains 308

5.5.a Convergence Theorems For Markov Chains 308

5.5.b Periodic Case Of Markov Chains 314

5.5.c Tail S-Field 316

5.6 General State Space 322

5.6.a Recurrence And Transience In A General State Space 325

5.6.b Stationary Measures In A General State Space 327

5.6.c Convergence Theorem In A General State Space 328

5.6.d Example: Gi/G/1 Queue 329

6 Ergodic Theorems 332

6.1 Definitions And Examples 332

6.2 Birkhoff's Ergodic Theorem 337

6.3 Recurrence 343

6.4 Mixing 347

6.5 Entropy 353

6.6 A Subadditive Ergodic Theorem 358

6.7 Applications 364

7 Brownian Motion 371

7.1 Definition And Construction 372

7.2 Markov Property, Blumenthal's 0-1 Law, [Blumenthal's 0-1 Law], [Markov Property] 378

7.3 Stopping Times, Strong Markov Property 384

7.4 Maxima And Zeros 389

7.5 Martingales For Brownian Motion 395

7.6 Donsker's Theorem 399

7.7 Clt For Dependent Variables 408

7.7.1 Clt For Martingales 408

7.7.2 Clt For Stationary Sequences 415

7.7.3 Clt For Mixing Properties 420

7.8 Empirical Distributions, Brownian Bridge, [Empirical Distribution], [Brownian Bridge] 425

7.9 Laws Of The Iterated Logarithm 431

8 Appendix: Measure Theory 437

8.1 Lebesgue-Stieltjes Measures 437

8.2 Carathéodory's Extension Theorem 444

8.3 Completion Of Measures 449

8.4 Integration 452

8.5 Properties Of The Integral 461

8.6 Product Measures, Fubini's Theorem, [Fubini's Theorem], [Product Measures] 466

8.7 Kolmogorov's Extension Theorem 471

8.8 Radon-Nikodym Theorem 473

8.9 Differentiating Under The Integral 479

Probability: Theory and Examples, 3Ed

Courses that use this book:

Probability: Theory And Examples, 3ed. Author(s): Richard Durrett. ISBN: 0534424414. Year: 2005. Publisher: Thomson Brooks/Cole.

Probability: Theory And Examples, 3ed.
Author(s): Richard Durrett.
ISBN: 0534424414.
Year: 2005.
Publisher: Thomson Brooks/Cole.