This book grew from a one-semester course offered for many years to a mixed audience of graduate and undergraduate students who have not had the luxury of taking a course in measure theory. The core of the book covers the basic topics of independence, conditioning, martingales, convergence in distribution, and Fourier transforms. In addition there are numerous sections treating topics traditionally thought of as more advanced, such as coupling and the KMT strong approximation, option pricing via the equivalent martingale measure, and the isoperimetric inequality for Gaussian processes. The book is not just a presentation of mathematical theory, but is also a discussion of why that theory takes its current form. It will be a secure starting point for anyone who needs to invoke rigorous probabilistic arguments and understand what they mean.
Author(s): David Pollard
Series: Cambridge Series in Statistical and Probabilistic Mathematics
Edition: 1
Publisher: Cambridge University Press
Year: 2001
Language: English
Pages: 366
Tags: Differential Equations;Applied;Mathematics;Science & Math;Probability & Statistics;Applied;Mathematics;Science & Math;Statistics;Mathematics;Science & Mathematics;New, Used & Rental Textbooks;Specialty Boutique
Contents ... 8
Preface ... 12
Chapter 1 Motivation ... 16
1. Why bother with measure theory? ... 16
2. The cost and benefit of rigor ... 18
3. Where to start: probabilities or expectations? ... 20
4. The de Finetti notation ... 22
5. Fair prices ... 26
6. Problems ... 28
7. Notes ... 29
Chapter 2 A modicum of measure theory ... 32
1. Measures and sigma-fields ... 32
2. Measurable functions ... 37
3. Integrals ... 41
4. Construction of integrals from measures ... 44
5. Limit theorems ... 46
6. Negligible sets ... 48
7. LP spaces ... 51
8. Uniform integrability ... 52
9. Image measures and distributions ... 54
10. Generating classes of sets ... 56
11. Generating classes of functions ... 58
12. Problems ... 60
Chapter 3 Densities and derivatives ... 68
1. Densities and absolute continuity ... 68
2. The Lebesgue decomposition ... 73
3. Distances and affinities between measures ... 74
4. The classical concept of absolute continuity ... 80
5. Vitali covering lemma ... 83
6. Densities as almost sure derivatives ... 85
7. Problems ... 86
Chapter 4 Product spaces and independence ... 92
1. Independence ... 92
2. Independence of sigma-fields ... 95
3. Construction of measures on a product space ... 98
4. Product measures ... 103
5. Beyond sigma-fi niteness ... 108
6. SLLN via blocking ... 110
7. SLLN for identically distributed summands ... 112
8. Infinite product spaces ... 114
9. Problems ... 117
10. Notes ... 123
Chapter 5 Conditioning ... 126
1. Conditional distributions: the elementary case ... 126
2. Conditional distributions: the general case ... 128
3. Integration and disintegration ... 131
4. Conditional densities ... 133
5. Invariance ... 136
6. Kolmogorov's abstract conditional expectation ... 138
7. Sufficiency ... 143
8. Problems ... 146
9. Notes ... 150
Chapter 6 Martingale et al. ... 153
1. What are they? ... 153
2. Stopping times ... 157
3. Convergence of positive supermartingales ... 162
4. Convergence of submartingales ... 166
5. Proof of the Krickeberg decomposition ... 167
6. Uniform integrability ... 168
7. Reversed martingales ... 170
8. Symmetry and exchangeability ... 174
9. Problems ... 177
10. Notes ... 181
Chapter 7 Convergence in distribution ... 184
1. Definition and consequences ... 184
2. Lindeberg's method for the central limit theorem ... 191
3. Multivariate limit theorems ... 196
4. Stochastic order symbols ... 197
5. Weakly convergent subsequences ... 199
6. Problems ... 201
Chapter 8 Fourier transforms ... 208
1. Definitions and basic properties ... 208
2. Inversion formula ... 210
3. A mystery? ... 213
4. Convergence in distribution ... 213
5. A martingale central limit theorem ... 215
6. Multivariate Fourier transforms ... 217
7. Cramer-Wold without Fourier transforms ... 218
8. The Levy-Cramer theorem ... 220
9. Problems ... 221
Chapter 9 Brownian motion ... 226
1. Prerequisites ... 226
2. Brownian motion and Wiener measure ... 228
3. Existence of Brownian motion ... 230
4. Finer properties of sample paths ... 232
5. Strong Markov property ... 234
6. Martingale characterizations of Brownian motion ... 237
7. Functionals of Brownian motion ... 241
8. Option pricing ... 243
9. Problems ... 245
Chapter 10 Representations and couplings ... 252
1. What is coupling? ... 252
2. Almost sure representations ... 254
3. Strassen's Theorem ... 257
4. The Yurinskii coupling ... 259
5. Quantile coupling of Binomial with normal ... 263
6. Haar coupling-the Hungarian construction ... 264
7. The Komlos-Major-Tusnady coupling ... 267
8. Problems ... 271
Chapter 11 Exponential tails and the law of the iterated logarithm ... 276
1. LIL for normal summands ... 276
2. LIL for bounded summands ... 279
3. Kolmogorov's exponential lower bound ... 281
4. Identically distributed summands ... 283
5. Problems ... 286
6. Notes ... 287
Chapter 12 Multivariate normal distributions ... 289
1. Introduction ... 289
2. Fernique's inequality ... 290
3. Proof of Fernique's inequality ... 291
4. Gaussian isoperimetric inequality ... 293
5. Proof of the isoperimetric inequality ... 295
6. Problems ... 300
7. Notes ... 302
Appendix A Measures and integrals ... 304
1. Measures and inner measure ... 304
2. Tightness ... 306
3. Countable additivity ... 307
4. Extension to the nc-closure ... 309
5. Lebesgue measure ... 310
6. Integral representations ... 311
7. Problems ... 315
8. Notes ... 315
Appendix B Hilbert spaces ... 316
1. Definitions ... 316
2. Orthogonal projections ... 317
3. Orthonormal bases ... 318
4. Series expansions of random processes ... 320
5. Problems ... 321
6. Notes ... 321
Appendix C Convexity ... 322
1. Convex sets and functions ... 322
2. One-sided derivatives ... 323
3. Integral representations ... 325
4. Relative interior of a convex set ... 327
5. Separation of convex sets by linear functionals ... 328
6. Problems ... 330
7. Notes ... 331
Appendix D Binomial and normal distributions ... 332
Appendix E Martingales in continuous time ... 344
1. Filtrations, sample paths, and stopping times ... 344
2. Preservation of martingale properties at stopping times ... 347
3. Supermartingales from their rational skeletons ... 349
4. The Brownian filtration ... 351
5. Problems ... 353
6. Notes ... 353
Appendix F Disintegration of measures ... 354
1. Representation of measures on product spaces ... 354
2. Disintegrations with respect to a measurable map ... 357
3. Problems ... 358
4. Notes ... 360
Index ... 362
