A Probability Path.pdf

# 446 index of probability space 64 autocovariance

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446 Index of probability space, 64 autocovariance function, 327 autoregression, 227 baby Skorohod theorem, 258, 259, 261,262 ballot problem, 439 Beppo Levi theorem, 165 Bernoulli random variable, 103, 163, 176,285 Bernstein polynomial, 176, 238 big block-little block method, 271 binomial distribution, 176, 282 birth process, 216 Bonferroni inequalities, 30 Borel sets, 16 comparing, 18 extended, 25 metric space, 18 open sets, 18 ways to generate, 17 Borel zero-one law, 103, 115, 219 Borel-Cantelli Lemma, 102, 163,204 partial converse, 236 branching process immigration, 435 bridge, 69 Cantor distribution, 244 Cauchy criterion, 171 Cauchy density, 126, 193, 286 Cauchy equation, 280 Cauchy in probability, 211 central limit theorem, 1, 253, 270, 282-284,287,294,302,313 DeMoivre-Laplace, 253 Liapunov condition, 319 Lindeberg condition, 314 Lindeberg-Feller, 314 m-dependence,270 proof, 312 central value, 162, 201 characteristic function, 293, 295 bilateral exponential density, 328 cauchy density, 328 continuity theorem, 304 derivatives and moments, 301 elementary properties, 296 expansions, 297 exponential density, 328 normal density, 299 selection theorem, 307, 309 triangle density, 329 uniform density, 328 uniqueness theorem, 302 Chebychev inequality, 130, 131, 176, 181,185,315 chf,295 circular Lebesgue measure, 86 closure, 11, 12, 35, 373, 374, 388 coincidences, 34 comparison lemma, 141 complement, 3 complete convergence, 250 complete market, 425,441 option pricing, 441 completion, 66 u-field, 66 composition, 77 conditional expectation, 2, 339 countable partitions, 341 definition, 340 densities, 344 discrete case, 343 existence, 340 independence,349 properties, 344 dominated convergence, 347 Fatou lemma, 346 Jensen inequality, 351 linearity, 344 modulus inequality, 345 monotone convergence, 346 monotonicity, 345 norm reducing, 352 product rule, 347 projections, 349 smoothing, 348 Radon-Nikodym derivative, 340 conditional probability, 341

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conditional variance, 363 consistency, 1 containment, 4 contingent claim, 425 attainable, 425 continuity, 31, 32 measurability, 80 measures, 31, 32 continuity theorem, 304, 326, 330 proof, 311 continuous functions, 79, 159 continuous mapping theorem, 261, 287 second,287 continuous maps, 174 continuous paths, 88 convergence Lp, 180 almost sure, 167 complete, 250 in distribution, 247, 251 in probability, 169 proper, 249 vague,249 weak, 249, 251 convergence in distribution, 247, 251 convergence in probability and almost sure convergence, 170, 171 and continuous maps, 174 and dominated convergence, 175 Cauchy criterion, 171 metrizable, 199 subsequence criterion, 172 convergence to types theorem, 274, 275, 279, 289, 290, 321, 329 converging together theorem first, 268 second,269,273 convolution, 154, 293 convolution transform, 288 coordinate map, 143, 162 correlation, 181 coupon collecting, 242 covariance, 128 crystal ball condition Index 447 and uniform integrability, 184 De Morgan's laws, 5 delta method, 261, 288 DeMoivre-Laplace central limit the- orem, 253 density, 135, 139 Bernoulli, 323 126,286,288,328 exponential, 285,287,288,323 extreme value, 279 gamma, 160, 163, 285, 325 normal, 162, 163, 284, 287,288, 299 Pareto, 286 Poisson, 288, 321 triangle, 329 uniform, 286,287,289,322 diagonalization, 307 discounted price process, 416 discrete uniform distribution, 266 distribution measure, 137 distribution function, 33, 38, 42, 61, 66,138,159,247 inverse, 61, 66 non-defective, 248 proper, 248 dominated convergence, 133, 157, 158, 160,161,163,164,175 and convergence in probability, 175 dominated family and uniform integrability, 183 dominating random variable, 132, 133 Doob decomposition, 360, 362
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