## A Modern Approach to Probability TheoryOverview This book is intended as a textbook in probability for graduate students in math ematics and related areas such as statistics, economics, physics, and operations research. Probability theory is a 'difficult' but productive marriage of mathemat ical abstraction and everyday intuition, and we have attempted to exhibit this fact. Thus we may appear at times to be obsessively careful in our presentation of the material, but our experience has shown that many students find them selves quite handicapped because they have never properly come to grips with the subtleties of the definitions and mathematical structures that form the foun dation of the field. Also, students may find many of the examples and problems to be computationally challenging, but it is our belief that one of the fascinat ing aspects of prob ability theory is its ability to say something concrete about the world around us, and we have done our best to coax the student into doing explicit calculations, often in the context of apparently elementary models. The practical applications of probability theory to various scientific fields are far-reaching, and a specialized treatment would be required to do justice to the interrelations between prob ability and any one of these areas. However, to give the reader a taste of the possibilities, we have included some examples, particularly from the field of statistics, such as order statistics, Dirichlet distri butions, and minimum variance unbiased estimation. |

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### Contents

Chapter 1 Probability Spaces | 3 |

12 Ingredients of probability spaces | 6 |

13 𝝈fields | 8 |

14 Borel 𝝈fields | 9 |

Chapter 2 Random Variables | 11 |

22 Rdvalued random variables | 14 |

23 Rvalued random variables | 18 |

24 Further examples | 20 |

191 Certain sequences of distributions on C0 1 | 368 |

192 The existence of and convergence to Wiener measure | 371 |

193 Some measurable functionals on C01 | 374 |

194 Drownian motion on 0 oc | 379 |

195 Filtrations and stopping times | 382 |

196 Brownian motion nitrations and stopping times | 385 |

197 Characteri2ation of Brownian motion | 389 |

198 Law of the Iterated Logarithm | 390 |

Chapter 3 Distribution Functions | 25 |

32 Examples of distributions | 29 |

33 Some descriptive terminology | 31 |

34 Distributions with densities | 35 |

35 Further examples | 37 |

36 Distribution functions for the extended real line | 39 |

Theory | 41 |

42 Linearity and positivity | 46 |

43 Monotone convergence | 49 |

44 Expectation of compositions | 51 |

45 The RiemannStieltjes integral and expectations | 53 |

Applications | 59 |

52 Mean vectors and covariance matrices | 66 |

53 Moments and the Jensen Inequality | 68 |

54 Probability generating functions | 70 |

55 Characterization of probability generating functions | 73 |

Chapter 6 Calculating Probabilities and Measures | 75 |

62 The BorelCantelli and KochenStone Lemmas | 77 |

63 Inclusionexclusion | 80 |

64 Finite and afinite measures | 81 |

Existence and Uniqueness | 85 |

72 Finitely additive functions defined on fields | 87 |

73 Existence extension and completion of measures | 90 |

74 Examples | 95 |

Chapter 8 Integration Theory | 101 |

82 Convergence theorems | 105 |

83 Probability measures and infinite measures compared | 110 |

84 Lebesgue integrals and RiemannStieltjes integrals | 111 |

85 Absolute continuity and densities | 115 |

86 Integration with respect to counting measure | 118 |

Chapter 9 Stochastic Independence | 121 |

finitely many factors | 127 |

93 The Fubini Theorem | 130 |

94 Expectations and independence | 133 |

95 Densities and independence | 134 |

infinitely many factors | 136 |

97 The BorelCantelli Lemma and independent sequences | 141 |

98 Order statistics | 143 |

99 Some new distributions involving independence | 145 |

Chapter 10 Sums of Independent Random Variables | 147 |

102 Multinomial distributions | 152 |

103 Probability generating functions and sums in Z | 153 |

104 Dirichlet distributions | 156 |

105 Random sums in various settings | 159 |

Chapter 11 Random Walk | 163 |

112 Definition and examples | 164 |

113 Filtrations and stopping times | 171 |

114 Stopping limes and random walks | 174 |

115 A hittingtime example | 175 |

116 Returns to 0 | 178 |

117 Random walks in various settings | 183 |

Chapter 12 Theorems of AS Convergence | 185 |

122 Laws of Large Numbers | 188 |

123 Applications | 193 |

124 01 laws In the Strong Law of Large Numbers we are interested in the event | 196 |

125 Random infinite series | 198 |

126 The Etemadi Lemma | 200 |

127 The Kolmogorov ThreeSeries Theorem | 202 |

128 The image of a random walk | 206 |

Chapter 13 Characteristic Functions | 209 |

132 The Parseval Relation and uniqueness | 211 |

133 Characteristic functions of convolutions | 214 |

134 Symmetrization | 217 |

135 Moment generating functions | 218 |

136 Moment theorems | 223 |

137 Inversion theorems | 226 |

138 Characteristic functions in Rd | 234 |

139 Normal distributions on ddimensional space | 237 |

1310 An application to random walks on Z | 238 |

1311 An application to the calculation of a sum | 239 |

Chapter 14 Convergence in Distribution on the Real Line | 243 |

142 Limit distributions for extreme values | 246 |

143 Relationships to other types of convergence | 249 |

144 Convergence conditions for sequences of distributions | 252 |

145 Sequences of distributions on R | 254 |

146 Relative sequential compactness | 255 |

147 The Continuity Theorem | 260 |

148 Scaling and centering of sequences of distributions | 263 |

149 Characteri2ation of moment generating functions | 266 |

1410 Characterization of characteristic functions | 269 |

Chapter 15 Distributional Limit Theorems for Partial Sums | 271 |

151 Infinite series of independent random variables | 272 |

152 The Law of Large Numbers revisited | 273 |

153 The Classical Central Limit Theorem | 275 |

154 The general setting for iid sequences | 277 |

155 Large deviations | 280 |

156 Local limit theorems | 282 |

Chapter 16 Infinitely Divisible Distributions as Limits | 289 |

162 Infinitely divisible distributions on R | 293 |

163 LévyKhinchin representations | 295 |

164 Infinitely divisible distributions on R+ | 300 |

165 Extension to R | 302 |

introduction | 303 |

167 lid triangular arrays | 306 |

168 Symmetric and nonnegative triangular arrays | 310 |

169 General triangular arrays | 313 |

Chapter 17 Stable Distributions as Limits | 323 |

171 Regular variation | 324 |

172 The stable distributions | 326 |

173 Domains of attraction | 332 |

174 Domains of strict attraction | 342 |

Chapter 18 Convergence in Distribution on Polish Spaces | 347 |

182 Definition of and criteria for convergence | 352 |

183 Relative sequential compactness | 355 |

184 Uniform tightness and the Prohorov Theorem | 357 |

185 Convergence in product spaces | 358 |

186 The Continuity Theorem for Rd | 361 |

187 The Prohorov metric | 363 |

Chapter 19 The Invariance Principle and Brownian Motion | 367 |

Chapter 20 Spaces of Random Variables | 395 |

202 The Hilbert space L2ΩFP | 397 |

203 The metric space L1ΩFP | 401 |

204 Best linear estimator | 402 |

Chapter 21 Conditional Probabilities | 403 |

212 Conditional distributions | 412 |

213 Conditional densities | 416 |

214 Existence and uniqueness of conditional distributions | 417 |

215 Conditional independence | 422 |

216 Conditional distributions of normal random vectors | 427 |

Chapter 22 Construction of Random Sequences | 429 |

222 Construction of exchangeable sequences | 433 |

223 Construction of Markov sequences | 436 |

224 Polya urns | 437 |

225 Coupon collecting | 439 |

Chapter 23 Conditional Expectations | 443 |

232 Conditional versions of unconditional theorems | 448 |

233 Formulas for conditional expectations | 451 |

234 Conditional variance | 453 |

Chapter 24 Martingales | 459 |

242 Examples | 461 |

243 Doob decomposition | 465 |

244 Transformations of submartingales | 466 |

optional sampling | 467 |

246 Applications of optional sampling | 473 |

247 Inequalities and convergence results | 477 |

248 Optimal strategy in Red and Black | 484 |

Chapter 25 Renewal Sequences | 489 |

251 Basic criterion | 490 |

252 Renewal measures and potential measures | 491 |

253 Examples | 494 |

a first step | 497 |

255 Delayed renewal sequences | 499 |

256 The Renewal Theorem | 502 |

257 Applications to random walks | 506 |

Chapter 26 Timehomogeneous Markov Sequences | 511 |

262 Examples | 515 |

263 Martingales and the strong Markov property | 520 |

264 Hitting times and return times | 522 |

265 Renewal theory and Markov sequences | 525 |

266 Irreducible Markov sequences | 527 |

267 Equilibrium distributions | 528 |

Chapter 27 Exchangeable Sequences | 533 |

272 Infinite exchangeable sequences | 539 |

273 Posterior distributions | 542 |

274 Generali2ation to Borel spaces | 544 |

275 Ferguson distributions and BlackwellMacQueen urns | 550 |

Chapter 28 Stationary Sequences | 553 |

282 Notation | 555 |

283 Examples | 556 |

284 The Birkhoff Ergodic Theorem | 558 |

285 Ergodicity | 561 |

286 The KingmanLiggett Subadditive Ergodic Theorem | 564 |

287 Spectral analysis of stationary sequences | 571 |

Chapter 29 Point Processes | 581 |

292 Intensity measures | 586 |

293 Poisson point processes | 587 |

294 Examples of Poisson point processes | 589 |

295 Probability generating functionals | 594 |

296 Operations on point processes | 597 |

297 Convergence in distribution for point processes | 598 |

Chapter 30 Levy Processes | 601 |

302 Definition of Levy process | 602 |

303 Construction of Levy processes | 605 |

304 Filtrations and stopping times | 610 |

305 Subordination | 611 |

306 Localtime processes and regenerative subsets of 0 oo | 612 |

307 Sample function properties of subordinators | 618 |

Chapter 31 Introduction to Markov Processes | 621 |

312 Markov strong Markov and Feller processes | 622 |

313 Infinitesimal generators | 628 |

314 The martingale problem We adapt the main definition of Chapter 24 to the continuoustime setting | 629 |

bounded rates | 632 |

unbounded rates | 636 |

317 Renewal theory for purejump Markov processes | 639 |

Chapter 32 Interacting Particle Systems | 641 |

322 The universal coupling | 644 |

323 Examples | 651 |

324 Equilibrium distributions | 655 |

325 Systems with attractive infinitesimal generators | 657 |

Chapter 33 DiIfusions and Stochastic Calculus | 661 |

332 The Itô integral | 663 |

333 Stochastic diIferentials and the Ito Lemma | 668 |

334 Autonomous stochastic differential equations | 672 |

335 Generators and the Dirichlet problem | 679 |

336 Diffusions in higher dimensions | 682 |

APPENDIX A Notation and Usage of Terms | 687 |

A 2 Usage Binary digits is used for what some call bits | 690 |

A3 Exercises on subtle distinctions | 692 |

APPENDIX B Metric Spaces | 693 |

B2 Sequences | 694 |

B3 Continuous functions | 695 |

APPENDIX C Topological Spaces | 697 |

C2 Compactification | 699 |

C3 Product topologies | 700 |

C5 Limits and continuous functions | 701 |

APPENDIX D RiemannStieltjes Integration | 703 |

D2 Relation to the Riemann integral | 705 |

D3 Change of variables | 706 |

D4 Integration by parts | 707 |

D5 Improper RiemannStieltjes integrals | 708 |

Appendix E Taylor Approximations CValued Logarithms | 709 |

E2 Complex exponentials and logarithms | 711 |

E3 Approximations of general Cvalued functions | 714 |

715 | |

APPENDIX G Comments and Credits | 723 |

737 | |

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### Common terms and phrases

a-field Borel space Borel subset bounded calculate Cauchy Chapter characteristic function compact conditional distribution constant convergence in distribution Convergence Theorem Corollary corresponding countable definition denote density distribution function distribution Q domain equal ergodic Example exchangeable sequence exists filtration Finetti measure finite measure formula Fubini Theorem generali2ation given Hint iid sequence inequality interval Large Numbers Lebesgue measure Lemma Let Q Levy measure Levy process limit Markov process Markov sequence martingale matrix measurable function measurable space metric space nonnegative obtain Poisson point processes Polish space positive integer preceding problem preceding proposition preceding theorem probability generating function probability measure probability space Problem 13 Problem 9 Prove the preceding R-valued random sequence random walk real number renewal sequence result Riemann-Stieltjes integral stationary sequence stochastic submartingale Suppose Theorem 12 topology transition operator uniformly unique valued random variables variance