Sampling without replacement 34 scheffe lemma 190 253

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sampling without replacement, 34 Scheffe lemma, 190, 253, 284, 287 and L 1 convergence, 190 Schwartz inequality, 186, 187, 196 second continuous mapping theorem, 287 second converging together theorem, 269,273 section function, 144, 146 set, 143, 145 selection theorem, 307, 309, 326 self-financing, 417 characterization, 417 semi-continuous function, 4, 87 semialgebra, 35, 43, 66, 144 field generated by, 45 intervals, 44 rectangles, 44 separating hyperplane theorem, 423 set difference, 3 set operations, 11 simple function, 84, 117-119, 136 expectation, 119 Skorohod inequality, 209, 210 Skorohodtheorem,258,259,261,262 Slutsky theorem, 268 spacings, 116, 285 St. Petersburg paradox, 240, 241 stationary process, 181 statistic, 171 Stirling formula, 323 CLT proof, 323 stochastic process, 88 stopped process, 88 stopping theorems, 392 stopping time, 363 comparison, 366 definition, 363 hitting time, 364 integrable, 407 preservation of process mean, 367 properties, 365 regular, 392 characterization, 394 criteria, 393, 394, 397 strong law oflarge numbers, 208, 213, 219,220 Kolmogorov, 220 strongly consistent, 171 structure, 35 minimal, 36 subadditivity measures, 31 submartingale, 386 subsequence criterion, 172 sums of independent random vari- ables, 209 superrnartingale, 366 Lebesgue decomposition, 382 positive, 366, 367 bounded,369 convergence,371,373 operations, 367 pasting, 367 stopped, 368 upcrossings, 369 stopped, 377 symmetric difference, 3 symmetrization, 232, 289 tail a-field, 107
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tail equivalence, 203, 204 tail event, 107 tail random variable, 107 three series theorem, 226, 237, 243, 244 ties, 89, 95, 97 tightness, 309 criteria, 310 trading strategy, 417 self-financing, 417 characterization, 417 transformation theorem, 135, 138, 293 transition function, 147 truncation, 203 type, 275 U-statistic, 438 UAN,315 uncorrelated, 130, 155, 165 uniform absolute continuity and uniform integrability, 184 uniform asymptotic negligibility, 315 uniform continuity, 67 uniform distribution, 266 uniform integrability, 182, 388 criteria, 183 uniform random number, 95, 98 dyadic expansion, 98 uniformly bounded, 157 Index 453 union, 3 uniqueness theorem, 302 upcrossings, 369 convergence, 369 upper semi-continuous, 4 vague convergence, 249 variance, 128, 155 vector space, 118 viable market, 420 Wald identity, 398, 405 weak L 1 convergence, 430 weak convergence, 249,251 equivalences, 263 metric, 285 weak law of large numbers, 204 applications, 239, 241 weakly consistent, 171 quantile estimation, 179 weakly stationary process, 327 Weierstrass approximation theorem Bernstein version, 176 zero-one law, 102 Borel, 103, 219 Borel-Cantelli Lemma, 102 Kolmogorov, 107, 108, 217
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