A Probability Path

Contents......Page 5
Preface......Page 11
1.1 Introduction......Page 13
1.2 Basic Set Theory......Page 14
1.2.1 Indicator functions 5......Page 17
1.3 Limits of Sets......Page 18
1.4 Monotone Sequences......Page 20
1.5 Set Operations and Closure......Page 23
1.5.1 Examples 13......Page 25
1.6 The sigma-field Generated by a Given Class C......Page 27
1.7 Borel Sets on the Real Line......Page 28
1.8 Comparing Borel Sets......Page 30
1.9 Exercises......Page 32
2.1 Basic Definitions and Properties......Page 41
2.2 More on Closure......Page 47
2.2.1 Dynkin's theorem......Page 48
2.2.2 Proof of Dynkin's theorem......Page 50
2.3 Two Constructions......Page 52
2.4 Constructions of Probability Spaces......Page 54
2.4.1 General Construction of a Probability Model......Page 55
2.4.2 Proof of the Second Extension Theorem......Page 61
2.5.1 Lebesgue Measure on (0,1]......Page 69
2.5.2 Construction of a Probability Measure on R with Given Distribution Function F(x)......Page 73
2.6 Exercises......Page 75
3.1 Inverse Maps......Page 83
3.2 Measurable Maps, Random Elements, Induced Probability Measures......Page 86
3.2.1 Composition 77......Page 89
3.2.2 Random Elements of Metric Spaces 78......Page 90
3.2.3 Measurability and Continuity 80......Page 92
3.2.4 Measurability and Limits 81......Page 93
3.3 sigma-Fields Generated by Maps......Page 95
3.4 Exercises......Page 97
4.1 Basic Definitions......Page 103
4.2 Independent Random Variables......Page 105
4.3.1 Records, Ranks, Renyi Theorem......Page 107
4.3.2 Dyadic Expansions of Uniform Random Numbers......Page 110
4.4 More on Independence: Groupings......Page 112
4.5.1 Borel-Cantelli Lemma......Page 114
4.5.2 Borel Zero-One Law......Page 115
4.5.3 Kolmogorov Zero-One Law......Page 119
4.6 Exercises......Page 122
5.1.1 Simple Functions......Page 129
5.1.2 Measurability and Simple Functions......Page 130
5.2.1 Expectation of Simple Functions......Page 131
5.2.2 Extension of the Definition......Page 134
5.2.3 Basic Properties of Expectation......Page 135
5.3 Limits and Integrals......Page 143
5.4 Indefinite Integrals......Page 146
5.5 The Transformation Theorem and Densities......Page 147
5.5.1 Expectation is Always an Integral on R......Page 149
5.6 The Riemann vs Lebesgue Integral......Page 151
5.7 Product Spaces, Independence, Fubini Theorem......Page 155
5.8 Probability Measures on Product Spaces......Page 159
5.9 Fubini's theorem......Page 161
5.10 Exercises......Page 167
6.1 Almost Sure Convergence......Page 179
6.2 Convergence in Probability......Page 181
6.2.1 Statistical Terminology......Page 182
6.3 Connections Between a.s. and i.p. Convergence......Page 183
6.4 Quantile Estimation......Page 190
6.5 L_p Convergence......Page 192
6.5.1 Uniform Integrability......Page 194
6.5.2 Interlude: A Review of Inequalities......Page 198
6.6 More on L_p Convergence......Page 201
6.7 Exercises......Page 207
7.1 Truncation and Equivalence......Page 215
7.2 A General Weak Law of Large Numbers......Page 216
7.3 Almost Sure Convergence of Sums of Independent Random Variables......Page 221
7.4 Strong Laws of Large Numbers......Page 225
7.4.1 Two Examples......Page 227
7.5 The Strong Law of Large Numbers for IID Sequences......Page 231
7.5.1 Two Applications of the SLLN......Page 234
7.6 The Kolmogorov Three Series Theorem......Page 238
7.6.1 Necessity of the Kolmogorov Three Series Theorem......Page 242
7.7 Exercises......Page 246
8.1 Basic Definitions......Page 259
8.2 Scheffe's lemma......Page 264
8.2.1 Scheffe's lemma and Order Statistics......Page 267
8.3 The Baby Skorohod Theorem......Page 270
8.3.1 The Delta Method......Page 273
8.4 Weak Convergence Equivalences; Portmanteau Theorem......Page 275
8.5 More Relations Among Modes of Convergence......Page 279
8.6 New Convergences from Old......Page 280
8.6.1 Example: The Central Limit Theorem for m-Dependent Random Variables......Page 282
8.7 The Convergence to Types Theorem......Page 286
8.7.1 Application of Convergence to Types: Limit Distributions for Extremes......Page 290
8.8 Exercises......Page 294
9 Characteristic Functions and the Central Limit Theorem......Page 305
9.1 Review of Moment Generating Functions and the Central Limit Theorem......Page 306
9.2 Characteristic Functions: Definition and First Properties......Page 307
9.3.1 Expansion of e^ix......Page 309
9.4 Moments and Derivatives......Page 313
9.5 Two Big Theorems: Uniqueness and Continuity......Page 314
9.6.1 The Selection Theorem......Page 319
9.6.2 Tightness, Relative Compactness, and Prohorov's theorem......Page 321
9.6.3 Proof of the Continuity Theorem......Page 323
9.7 The Classical CLT for iid Random Variables......Page 324
9.8 The Lindeberg-Feller CLT......Page 326
9.9 Exercises......Page 333
10.1 Prelude to Conditional Expectation: The Radon-Nikodym Theorem......Page 345
10.2 Definition of Conditional Expectation......Page 351
10.3 Properties of Conditional Expectation......Page 356
10.4 Martingales......Page 365
10.5 Examples of Martingales......Page 368
10.6.1 Doob's Decomposition......Page 372
10.7 Stopping Times......Page 375
10.8 Positive Super Martingales......Page 378
10.8.1 Operations on Supermartingales......Page 379
10.8.3 Boundedness Properties......Page 381
10.8.4 Convergence of Positive Super Martingales......Page 383
10.8.5 Closure......Page 386
10.8.6 Stopping Supermartingales......Page 389
10.9.1 Gambler's Ruin......Page 391
10.9.2 Branching Processes......Page 392
10.9.3 Some Differentiation Theory......Page 394
10.10.1 Krickeberg Decomposition......Page 398
10.10.2 Doob's (Sub)martingale Convergence Theorem......Page 399
10.11 Regularity and Closure......Page 400
10.12 Regularity and Stopping......Page 402
10.13 Stopping Theorems......Page 404
10.14 Wald's Identity and Random Walks......Page 410
10.14.1 The Basic Martingales......Page 412
10.14.2 Regular Stopping Times......Page 414
10.14.3 Examples of Integrable Stopping Times......Page 419
10.14.4 The Simple Random Walk......Page 421
10.15 Reversed Martingales......Page 424
10.16.1 A Simple Market Model......Page 428
10.16.2 Admissible Strategies and Arbitrage......Page 431
10.16.3 Arbitrage and Martingales......Page 432
10.16.4 Complete Markets......Page 437
10.16.5 Option Pricing......Page 440
10.17 Exercises......Page 441
References......Page 455
Index......Page 457