Probability: theory and examples
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge Univ. Press
2010
|
| Ausgabe: | 4. ed. |
| Schriftenreihe: | Cambridge series in statistical and probabilistic mathematics
[31] |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | X, 428 S. graph. Darst. |
| ISBN: | 9780521765398 |
Internformat
MARC
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Datensatz im Suchindex
| _version_ | 1804143327015075840 |
|---|---|
| adam_text | Titel: Probability
Autor: Durrett, Richard
Jahr: 2010
Contents
Preface page ix
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
vi Contents
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 (Erdös-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
Contents vii
6.7 Periodicity, Tail s-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
333
338
342
347
353
353
359
365
370
370
371
375
376
380
382
391
396
401
401
407
410
412
416
419
425
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.I Carathéodory s Extension Theorem
A.2 Which Sets Are Measurable?
A.3 Kolmogorov s Extension Theorem
A.4 Radon-Nikodym Theorem
A.5 Differentiating under the Integral
References
Index
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| spellingShingle | Durrett, Richard 1951- Probability theory and examples Cambridge series in statistical and probabilistic mathematics Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
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