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Contents
1 Measure Theory 1
1.1 Probability Spaces . . . . . . . . . . . 1
1.2 Distributions . . . . . . . .. . . . . . . . . 8
1.3 Random Variables . . . . . . . . . . . . 12
1.4 Integration . . . . . . . . . . . . . . . 15
1.5 Properties of the Integral . . . . . . . . . 21
1.6 Expected Value . . . . . . . . . . . . 24
1.6.1 Inequalities . . . . . . . . .. . . . 24
1.6.2 Integration to the Limit . . . . . . 25
1.6.3 Computing Expected Values . . . . . . 27
1.7 Product Measures, Fubini’s Theorem . . . .. 31
2 Laws of Large Numbers 37
2.1 Independence . . . . . . . . . . . . . 37
2.1.1 Sufficient Conditions for Independence . . . . . . . 38
2.1.2 Independence, Distribution, and Expectation . . .. . . 41
2.1.3 Sums of Independent Random Variables . . . . . . . . 42
2.1.4 Constructing Independent Random Variables . . . . 45
2.2 Weak Laws of Large Numbers . . . . . . . . . . 47
2.2.1
L2 Weak Laws . . . . . . . . . . . 472.2.2 Triangular Arrays . .. . . . . . . 50
2.2.3 Truncation . . . . . . . . . . . . . 52
2.3 Borel-Cantelli Lemmas . . . . . . . . . . 56
2.4 Strong Law of Large Numbers . . . . . . . . . 63
2.5 Convergence of Random Series* . . . .. . . . 68
2.5.1 Rates of Convergence . . .. . . . . . 71
2.5.2 Infinite Mean . . . . . . . . . . . . . 73
2.6 Large Deviations* . . . . . . . . . . 75
3 Central Limit Theorems 81
3.1 The De Moivre-Laplace Theorem . . . . . . . 81
3.2 Weak Convergence . . . . .. . . . . 83
3.2.1 Examples . . . . . . . . . . . 83
3.2.2 Theory . . . . . . . . . . . . . . 86
3.3 Characteristic Functions . . . . . . . . . . 91
3.3.1 Definition, Inversion Formula . . . . . . . . . 91
3.3.2 Weak Convergence . . . . . . . . . . . 97
3.3.3 Moments and Derivatives . . . . . . . 98
3.3.4 Polya’s Criterion* . . . . . . . . 101
iii
iv
CONTENTS3.3.5 The Moment Problem* . . . . .. . . . 103
3.4 Central Limit Theorems . . . . 106
3.4.1 i.i.d. Sequences . . . . 106
3.4.2 Triangular Arrays . . . . .. . . 110
3.4.3 Prime Divisors (Erd¨os-Kac)* . . . .. . 114
3.4.4 Rates of Convergence (Berry-Esseen)* . . .118
3.5 Local Limit Theorems* . . . . 121
3.6 Poisson Convergence . . . .. 126
3.6.1 The Basic Limit Theorem . . . . . . 126
3.6.2 Two Examples with Dependence . . .. 130
3.6.3 Poisson Processes . . .. 132
3.7 Stable Laws* . . . . . . . 135
3.8 Infinitely Divisible Distributions* . . .. 144
3.9 Limit Theorems in R
d . . . . . . 1474 Random Walks 153
4.1 Stopping Times . . . . . 153
4.2 Recurrence . . . . . . . . 162
4.3 Visits to 0, Arcsine Laws* . . .. 172
4.4 Renewal Theory* . . . .. . 177
5 Martingales 189
5.1 Conditional Expectation . . . 189
5.1.1 Examples . . . . . 191
5.1.2 Properties . . . 193
5.1.3 Regular Conditional Probabilities* . . . . 197
5.2 Martingales, Almost Sure Convergence . . . . 198
5.3 Examples . . . . . . . 204
5.3.1 Bounded Increments . . . . . 204
5.3.2 Polya’s Urn Scheme . . . . . . 205
5.3.3 Radon-Nikodym Derivatives . . . . . 206
5.3.4 Branching Processes . . . . . . 209
5.4 Doob’s Inequality, Convergence in
Lp . . . . . 2125.4.1 Square Integrable Martingales* . .. . . 216
5.5 Uniform Integrability, Convergence in
L1 . . . . 2205.6 Backwards Martingales . . . . . . 225
5.7 Optional Stopping Theorems . . . . . 229
6 Markov Chains 233
6.1 Definitions . . . . . . . 233
6.2 Examples . . . . . . . . 236
6.3 Extensions of the Markov Property . . . . . 240
6.4 Recurrence and Transience . . . . 245
6.5 Stationary Measures . . . .. . . 252
6.6 Asymptotic Behavior . . . . . 261
6.7 Periodicity, Tail
-field* . . . . . 2666.8 General State Space* . . . . . . 270
6.8.1 Recurrence and Transience . . . 273
6.8.2 Stationary Measures . . . 274
6.8.3 Convergence Theorem . . 275
6.8.4 GI/G/1 queue . . . . 276
CONTENTS
v7 Ergodic Theorems 279
7.1 Definitions and Examples . . . . 279
7.2 Birkhoff’s Ergodic Theorem . . . 283
7.3 Recurrence . . . . . . . 287
7.4 A Subadditive Ergodic Theorem* . 290
7.5 Applications* . . . . . 294
8 Brownian Motion 301
8.1 Definition and Construction . . .. 301
8.2 Markov Property, Blumenthal’s 0-1 Law . . 307
8.3 Stopping Times, Strong Markov Property . 312
8.4 Path Properites . . . .. 315
8.4.1 Zeros of Brownian Motion . . . . . 316
8.4.2 Hitting times . . . .. . 316
8.4.3 L´evy’s Modulus of Continuity . . . . . 319
8.5 Martingales . . . . . 320
8.5.1 Multidimensional Brownian Motion . . . . 324
8.6 Itˆo’s formula* . . . . . . 327
8.7 Donsker’s Theorem . . . . .333
8.8 CLT’s for Martingales* . . .340
8.9 Empirical Distributions, Brownian Bridge . . . . 346
8.10 Weak convergence* . . . . 351
8.10.1 The space
C . . . . . . . 3518.10.2 The Space D . . . . . . . 353
8.11 Laws of the Iterated Logarithm* . . . 355
A Measure Theory Details 359
A.1 Carathe
’eodory’s Extension Theorem . . . 359
A.2 Which Sets Are Measurable? . . 364
A.3 Kolmogorov’s Extension Theorem . . . 366
A.4 Radon-Nikodym Theorem . . . . 368
A.5 Differentiating under the Integral . . . . . 371
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