Probability

This is the first of four books I have written; the one I worked the hardest on; and the one I am fondest of. It marked my goodbye to mathematics and probability theory. About the time the book was written, I left UCLA to go into the world of applied statistics and computing as a full-time freelance consultant.The book went out of print well over ten years ago, but before it did a generation of statisticians, engineers, and mathematicians learned graduate probability theory from its pages. Since the book became unavailable, I have received many calls asking where it could be bought and then for permission to copy part or all of it for use in graduate probability courses.These reminders that the book was not forgotten saddened me and I was delighted when SIAM offered to republish it in their Classics Series. The present edition is the same as the original except for the correction of a few misprints and errors, mainly minor.After the book was out for a few years it became commonplace for a younger participant at some professional meeting to lean over toward me and confide that he or she had studied probability out of my book. Lately, this has become rarer and the confiders older. With republication. I hope that the age and frequency trends will reverse direction. Front Cover......Page 1 Series......Page 3 Title......Page 6 Copyright......Page 7 Preface to the Classic Edition......Page 8 Preface......Page 10 Contents......Page 12 2 The "law of averages"......Page 18 3 The bell-shaped curve enters (fluctuation theory)......Page 24 4 Strong form of the "law of averages"......Page 28 5 An analytic model for coin-tossing......Page 32 6 Conclusions......Page 34 1 Introduction......Page 36 2 Random vectors......Page 37 3 The distribution of processes......Page 38 4 Extension in sequence space......Page 40 5 Distribution functions......Page 42 6 Random variables......Page 46 7 Expectations of random variables......Page 48 8 Convergence of random variables......Page 50 1 Basic definitions and results......Page 53 2 Tail events and the Kolmogorov zero-one law......Page 57 3 The Borel-Cantelli lemma......Page 58 4 The random signs problem......Page 62 5 The law of pure types......Page 66 6 The law of large numbers for independent random variables......Page 68 7 Recurrence of sums......Page 70 8 Stopping times and equidistribution of sums......Page 75 9 Hewitt-Savage zero-one law......Page 80 1 Introduction......Page 84 2 A more general conditional expectation......Page 90 3 Regular conditional probabilities and distributions......Page 94 1 Gambling and gambling systems......Page 99 2 Definitions of martingales and submartingales......Page 100 3 The optional sampling theorem......Page 101 4 The martingale convergence theorem......Page 106 5 Further martingale theorems......Page 108 6 Stopping times......Page 112 7 Stopping rules......Page 115 8 Back to gambling......Page 118 1 Introduction and definitions......Page 121 2 Measure-preserving transformations......Page 123 3 Invariant sets and ergodicity......Page 125 4 Invariant random variables......Page 129 5 The ergodic theorem......Page 130 6 Converses and corollaries......Page 133 7 Back to stationary processes......Page 135 8 An application......Page 137 9 Recurrence times......Page 139 10 Stationary point processes......Page 142 1 Definitions......Page 146 2 Asymptotic stationarity......Page 150 3 Closed sets, indecomposability, ergodicity......Page 152 4 The countable case......Page 154 5 The renewal process of a state......Page 155 6 Group properties of states......Page 158 7 Stationary initial distributions......Page 160 8 Some examples......Page 162 9 The convergence theorem......Page 167 10 The backward method......Page 170 1 Introduction......Page 176 2 The compactness of distribution functions......Page 177 3 Integrals and D-convergence......Page 180 4 Classes of functions that separate......Page 182 5 Translation into random-variable terms......Page 183 6 An application of the foregoing......Page 184 7 Characteristic functions and the continuity theorem......Page 187 9 Characteristic functions and independence......Page 192 10 Fourier inversion formulas......Page 194 11 More on characteristic functions......Page 196 13 Other separating function classes......Page 199 2 Why normal?......Page 202 3 The nonidentically distributed case......Page 203 4 The Poisson convergence......Page 205 5 The infinitely divisible laws......Page 207 6 The generalized limit problem......Page 212 7 Uniqueness of representation and convergence......Page 213 8 The stable laws......Page 216 9 The form of the stable laws......Page 217 10 The computation of the stable characteristic functions......Page 221 11 The domain of attraction of a stable law......Page 224 12 A coin-tossing example......Page 230 13 The domain of attraction of the normal law......Page 231 2 The tools......Page 233 3 The renewal theorem......Page 235 4 A local central limit theorem......Page 241 5 Applying a Tauberian theorem......Page 244 6 Occupation times......Page 246 1 Introduction......Page 250 2 Properties of N_k......Page 251 3 The multidimensional central limit theorem......Page 254 4 The joint normal distribution......Page 255 5 Stationary Gaussian process......Page 258 6 Spectral representation of stationary Gaussian processes......Page 259 7 Other problems......Page 263 1 Introduction......Page 265 3 Definitions and existence......Page 268 4 Beyond the Kolmogorov extension......Page 271 5 Extension by continuity......Page 272 6 Continuity of Brownian motion......Page 274 7 An alternative definition......Page 276 8 Variation and differentiability......Page 278 9 Law of the iterated logarithm......Page 280 10 Behavior at t = x......Page 282 11 The zeros of X(t)......Page 284 12 The strong Markov property......Page 285 1 Introduction......Page 289 2 The first-exit distribution......Page 290 3 Representation of sums......Page 293 4 Convergence of sample paths of sums to Brownian motion paths......Page 295 5 An invariance principle......Page 298 6 The Kolmogorov-Smirnov statistics......Page 300 7 More on first-exit distributions......Page 304 8 The law of the iterated logarithm......Page 308 9 A more general invariance theorem......Page 310 2 The extension to smooth versions......Page 315 3 Continuous parameter martingales......Page 317 4 Processes with stationary, independent increments......Page 320 5 Path properties......Page 323 6 The Poisson process......Page 325 7 Jump processes......Page 327 8 Limits of jump processes......Page 329 9 Examples......Page 333 10 A remark on a general decomposition......Page 335 1 Introduction and definitions......Page 336 2 Regular transition probabilities......Page 337 3 Stationary transition probabilities......Page 339 4 Infinitesimal conditions......Page 341 5 Pure jump processes......Page 345 6 Construction of jump processes......Page 349 7 Explosions......Page 353 8 Nonuniqueness and boundary conditions......Page 356 9 Resolvent and uniqueness......Page 357 10 Asymptotic stationarity......Page 361 1 The Ornstein-Uhlenbeck process......Page 364 2 Processes that are locally Brownian......Page 368 3 Brownian motion with boundaries......Page 369 4 Feller processes......Page 373 5 The natural scale......Page 375 6 Speed measure......Page 379 7 Boundaries......Page 382 8 Construction of Feller processes......Page 387 9 The characteristic operator......Page 392 10 Uniqueness......Page 396 11 (φ_+(x) and (φ_-(x))......Page 400 Appendix: On Measure and Function Theory......Page 408 Bibliography......Page 422 Index......Page 429