Index. Ballot problem, 250 Banach space, 111

Transcription

1 d measurable function, 15 Abbott, 163 Abel, 114, 119 Absolutely continuous distribution function, 27, 270 set function, 203, 205 random variables, 27 Additive (set function), 18 a-, 19 see also Subadditive set function Algebra, 6 degenerate a-, (of events), 238 product a-, 8 semi-,20 a-,6 see also a-algebra generated by; a-algebra of permutable events tail a-, 63 Almost certainly (surely), 20 Almost everywhere, 171 Anscombe, 340, 353 At random, 56 Austin, 408, 442 Ballot problem, 250 Banach space, 111 Barndorff-Nielsen, 65, 83 Baum, 83, 135, 163,401 Baum-Katz-Spitzer theorem, 135 converse, 199 Bayes theorem, 228 Bernoulli, 53, 75 trials with parameter p, trials with success probability p, 56 weak law of large numbers, 39 Bernstein, 53, 353 inequality, 111 polynomial, 42 Berry, 318,322, 353 -Esseen theorem, 322, 337 Binomial distribution function, 31,294 negative, 38, 60 random variable, 31, 50, 56 Bivariate Poisson distribution, 191 Blackwell, 156,437,442 Blum, 341, 353 Bochner theorem, 308 Bonferroni inequalities, 38 Borel, 53, 75 -Cantelli lemma, 42, 44,101, 102 -Cantelli theorem, 61, 96,

2 480 Borel (cant.) (measurable) function, 14 line, 11 set, 11 space, 11, 186 strong law of large numbers, 42 zero-one criterion, 61 Bounded in probability, 273 Branching process, 258 Bray, 274,276,311 Breiman, 258, 268 Brown, B., 161, 163,428,442 Brunk, 363, 401 -Chung strong law of large numbers, 363 Buhlman, 268 Burkholder, 163,414,422,425,442 -Davis-Gundy inequality, 425 inequality, 414 Cantelli, 42, 44, 61, 83, 96, 258, 284, 311 Carleman criterion, 303 Cauchy convergence criterion, 99 distribution function, 294 Central limit theorem, 48, 313, 345 De Moivre-Laplace, 46, 47 Doeblin-Anscombe, 340 for martingales, 327, 336, 345 for Poisson random variables, 52 for sums of interchangeable random variables, 328, 329 for V-statistics, 326 Liapounov, 316 Lindeberg-Feller, 314 Characteristic exponent, 469 Characteristic function, 286 r-analytic, 301 entire, 301 Chernoff, 281, 311, 353 Chow, 111, 163, 221, 268, 286, 311, 401,402,442 Chung, 70, 71, 83, 98, 111, 123, 130, 137,154,163,268,311,353, 402,477 Class, 1 J..-,7 monotone, 6 1t-,7 Complement, 4 Complete compactness, 282 convergence of distribution functions, 271 convergence of random variables, 43 Completion (of), 25, 236 Conditional density, 224 distribution function, 225 expectation, 210, 212, 213, independence, probability, 59, 222, 223 probability measure (regular conditional probability), 223 see also Regular conditional distribution, 225 Consistent family of distribution functions, 195 Continuity point, 175, 271 Convergence almost certainly (a.c.) or almost surely (a.s.), 43, 66 a.c. unconditionally, 121 almost everywhere (a.e.), 172, 173 complete (for distribution functions), 271, 309 complete (for random variables), 44 in distribution (law), 272 in mean of order p, 95, 99, 102, 104,109 in measure, 172 in probability, 43, 66, moment, 317 weak,271 Convex function, , 110 inequality for martingales, 421, 422,425

3 481 Convolution, 189 Coordinate random variable, 57, 187, 196 Copies (of a stopping time), 141, 142, 146, 147, 150, 155 Correlation coefficient, 106 Countable set, 1 Counting measure, 24 Covariance, 109 Cramer, 273, 311,477 - Levy theorem, 305 Davis, 418, 422, 425, 442 Defective stopping time, 138 de Finetti, 33, 53, 234, 268 Degenerate distribution function, 31, 70, 294 a-algebra (of events), 238 random variable, 31,64 V-statistic, 259 Delayed sum, 135 De Moivre, 53, 75 -Laplace (central limit) theorem, Density (function), 27,31,46,206 Discrete distribution function, 27 random variable, 27 Disjoint class, 4 Distinguished logarithm, 446 nth root, 446 Distribution function (d.f.), 25, 26, 28,30,270,271 absolutely continuous, 270 associated, 370 binomial, 31 bivariate Poisson, 191 Cauchy, 294 conditional, 225 degenerate, 31,270,294 discrete, 27 exponential, 60 gamma, 294 geometric, 60 hypergeometric, 39 infinitely divisible, 445 inverse triangular, 294 joint, 26, 187 joint normal, 208, 221, 228 marginal, 309 multinomial, 191 n-dimensional, 187 negative binomial, 38, 60 normal, 31, 53, 294 Poisson, 31,294 positive normal, 78, 343 positive stable, 343 sample (empirical), 284 singular, 270 stable, 468 symmetric Bernoulli, 294 triangular, 294 Doeblin, 340, 353 -Anscombe central limit theorem, 340 Doob, 29, 83, 111, 164,209,223, 225,239,245,268,291,311, 353,402,442 maximal inequalities, 255 upcrossing inequality, 405 Domain of attraction, 477 Dubins, 254, 268, Dvoretzky, 339, 348, 352, 353,441 Dynkin, 29, 268 Egorov, D. H., theorem, 75 Egorov, V. A., 402 Elementary function, 85 Equicontinuous, 208 Equivalence of measures, 207 of sequences of random variables, 116 Erdos, 135,342,353 Erickson, 164, 183 Esseen, 318, 322, 353 Essential supremum, 202 Etemadi, 131, 132 Exchangeable (see interchangeable) random variables

4 482 Event, 19 Expectation (mean), 84 existence of, 84 Exponential distribution, 60 random variable, 60, 65 Extension of a sequence, 90 of a set function, 20, 165 Factor closed, 300 Fatou lemma, 95, 174 for conditional expectations, 216 extended, 218 Feller, 53, 61, 70, 83, 126, 128, 137, 164,311,313,314,353,402 -Chung lemma, 70 weak law of large numbers, 128 Fictitious random variable, 272 Finite measure space, 23 partition, 18 permutation, 232 real line, 11 set, 2 set function, 18 stopping time (rule, variable), 138, 240 Finitely additive set function, 18 First passage time, 141 Frechet, 283, 311 -Shohat theorem, 283 Freedman, 254,268,442 Friedman, 325 Fubini theorem, 186, 191,213 Fuchs, 98, 111, 154, 163 Function d-measurable,15 absolutely continuous, 27 additive, 18 Borel (measurable), 14 convex, density, 27, 31,46,206 discrete distribution, 27 distribution, 26 elementary, 85 finite set, 18 integrable, 84, 89, 172 joint distribution, 26, 187 left continuous, 25, 26 moment generating, 110 monotone set, 19 probability density, 27 set, 18 (i-additive (countably additive) set, 19 (i-finite set, 18 simple, 18 subadditive set, 19 subtractive set, 24 tail,64 Gambler's ruin problem, 81, 250 Gamma distribution, 294 Garsia, 425, 442 Geometric distribution, 60, 91 Glivenko, 284, 311 -Cantelli theorem, 284, 311 Gnedenko, 312,402,477 Gundy, 157, 164,422,425,442 Haag, 33, 53 Hajek, 255, 268 -Renyi inequality, 255 Halmos,29, 111, 177,209,222,268 Hanson, 353 Hardy, 53, 103, 111,209,297,312, 368,372 Harris, 258 Hartman, 374, 378,402 - Wintner law of the iterated logarithm, 382 Hausdorff, 29, 368, 372 Helly, 274, 312 -Bray lemma, 274 -Bray theorem, 276 Hewitt, 238, 268 Heyde, 137,164, 346, 353 Hoeffding, 268 decomposition, 261

5 483 Holder inequality, 104, 109, 174,227 Hsu,44,53,135,393,402 Hypergeometric distribution, 39 Identifiable, 294 Independent classes, 54 conditionally, events, 54 families of random variables, 55 identically distributed (i.i.d.) random variables, 55 Indicator, 4 Induced measure, 25, 176 Infinite dimensional product measurable space, 10 measure space Infinitely often (i.o.), 2 divisible (distribution), 445 Inequality Burkholder, 414 Burkholder-Davis-Gundy, 422, 425 Doob maximal, 255 Doob upcrossing, Hajek-Renyi,255 Holder, 104, 109, 174, 227 Jensen, 104,217 Khintchine, 384 Kolmogorov, 133, 255 Levy, 72 Marcinkiewicz-Zygmund, 386 Markov,86,89,173 Minkowski, 110, 183 Ottaviani,75 Schwarz, 105 Tchebychev, 40, 106 Young, 184,424 Integrable, 84, 93, 172 function, 172 uniformly, 93 Integral, 84, 171 indefinite, 92, 93 Lebesgue, 176 Lebesgue-Stieltjes, 175, 180 Riemann, 178 Riemann-Stieitjes, Interchangeable (exchangeable) events, 33 random variables, 191, , 238,241 Inverse triangular distribution, 294 Inversion formula, 287 Jensen inequality, 104 for conditional expectations, 217 Joint distribution function, 26, 187 normal distribution, 208 probability density function, 55 Jordan decomposition, 208 Kac, 342, 353 Katz, 83, 135, 163, 325, 401 Kawata, 164 Kemperman, 267 Kendall, 228, 268 Kesten, 156, 164,214 Khintchine, 113, 164, 368, 372, 402, 451 inequality, Kolmogorov convergence theorem, 113 Kiefer, 397, 402 Kingman, 402 Klass, 137, 138, 164, 269 Knopp, 164,353 Kochen, 102, 112 Kolmogorov, 83,113,124,164,208, 312,368,372,402,445,452, 477 consistency theorem, 195 inequality, 133, 255 law of the iterated logarithm, 373 strong law of large numbers, 125 three series theorem, 117 zero-one law, 64 Komlos, 137

6 484 Koopmans, 325 Krickeberg, 269 Kronecker lemma, 114 Lai,402 Laplace, 46, 47, 53, 75 Law of the iterated logarithm (LIL) independent random variables, V-statistics, 382 Hartman-Wintner, 382 Kolmogorov, 373 Law of large numbers, Strong (SLLN), Weak (WLLN) definition, 124 Bernoulli (weak), 39 Borel (strong), 42 Brunk-Chung (strong), 363 Etemadi, 132 for arrays, 138,393 for interchangeable random variables, 235 for V-statistics, 263 Feller (weak), 128 generalized SLLN, , generalized WLLN, ,467 Kolmogorov (strong), 125 Marcinkiewicz-Zygmund (strong), 125 Lebesgue decomposition theorem, 204 dominated convergence theorem, 100,174 for conditional expectations, 216 integral, 176 measurable set, 170 measure, 170 measure space, 170 monotone convergence theorem, 86,90,95,173 for conditional expectations, 216 -Stieltjes measure, 170, 188 -Stieltjes measure space, 170, 175, 188 see also Non-Lebesgue measurable set, 171 Left continuous, 25, 168 Levy, 71, 72,83,164,239,245,269, 305,312,353,451,477 class.p (of distributions), 468 concentration function, 286 continuity theorem, 289 distance, 278 inequality, 72 inversion formula, Khintchine representation, 451 theorem, 72 Liapounov, 105, 112 central limit theorem, 316 Likelihood ratio, 257 Lindeberg, 313, 353 condition, 313, 317 -Feller central limit theorem, 314 Linear Borel set, 11 Littlewood, 111, 209, 368, 372 Loeve, 83, 112, 117, 124, 164,209, 269,312,402,477.P p random variable, 95.P p space, 95 Lukacs, 312 Marcinkiewicz, 118, 121, 125, 164, 386,387,402 -Zygmund inequality, 387 -Zygmund strong law of large numbers, 125, 138, 263, 393 Marginal distribution, 309 Markov chain, 238 inequality, 86, 89, 173 Martingale, 239 central limit theorems, 336, 345, 346,351 convergence theorems, 247, 248, , 411 differences, 241, 336

7 485 inequalities, 255, 256, 409, 414, 418,425,427,430,434 moments of stopped (martingales), 250, 253, 431 reversed (downward), 241 Wald equation for, ,415 Match, 38 Maximal inequalities, 255, 256, 368 McShane, 209 McLeish, 348, 349, 353 Mean convergence criterion, 99 Measurable cover, 228 function, 14 rectangle, 9 set, 8 see also d -measurable function; Lebesgue or v-measurable set Measure, 19 complete, 168 conditional probability, 222 convergence in, 172 counting, 24 extension, 165 finite, 23 induced, 25 Lebesgue, (and measure space), 170 Lebesgue-Stieltjes, (and measure space), 170 n-dimensional Lebesgue-Stieltjes, 188 outer, 168 product, 184 restriction ofa, 165 a-finite, 165 signed, 208 space, 19 see also Infinite (and n-) dimensional product measure space, 185, 193 Median, 71, 109 Minkowski inequality, 110, 183 Mixture, 190,286,292,294,345,453 Mogyorodi, 341, 353 Moment, 105 convergence, 277, 317,442 generating function, 110,281 of randomly stopped sums, 250, 253,431,432 Monotone class, 6 convergence theorem, 86, 90, 95, 173 convergence theorem for conditional expectations, 216 sequence of sets, 3 set function, 19 system 15 Monroe, 209 Multinomial distribution, 191 Nagaev, 366,402 n-dimensional Borel measure space, 186 distribution function, 187 Negative binomial distribution 38, 60 part, 15 Neveu, 442 Newman, 61 Nikodym, 204, 205 Non-Lebesgue measurable set, 171 Normal distribution, 31, 294 positive, 78, 343 Normal random variable, 31 v-measurable set, 168 Null event, 20 Number of upcrossings, Optimal stopping rule, 160 Ornstein, 130, 154, 163 Ottaviani, 305 inequality, 75 Outer measure, 168 p-norm,i04 Panzone, 405, 443 Parameter, 31 Periodic, 297

8 486 Permutable events, 232 Poincare formula, 33 Point of continuity, 175,271 of increase, 28,271 Poisson distribution, 31, 52,294 random variable, 31, 65 theorem, 32 Polya, 98, 112,299,312 Positive definite, 308 normal distribution, 78, 343 part, 15 stable distribution, 343 Probability, 19 conditional, 59, 222, 223 density function, 27 space, 19 success probability, 56 Product measurable space, 8 measure, 184 measure space, 184, 185 a-algebra, 8 space, 8 Prohorov, 402 Rademacher functions, 201 Radon - Nikodym derivative, Nikodym theorem, 204 Raikov, 312,402,467 -Ottaviani theorem, 308 Random allocation of balls into cells, 56, 57, vector, 26 walk, 76, see also Simple random walk Random variable, 20 absolutely continuous, 27 binomial,31 coordinate, 57, 196,234,235 degenerate, 31 discrete, 27 exponential, 65 fictitious, 272 independent, identically distributed (i.i.d.), 55 interchangeable (exchangeable), 191, ,238,241 2 p -, 95 normal, 31, 65 Poisson, 31, 65 symmetric, 72, 74 symmetrized, 197 Real line, 11 Rectangle, 9 Recurrent, 97, 98, 130, 163 Regular conditional distribution, 225 Renyi, 52, 53, 268, 353 Renewal theorem (elementary), Restriction, 20, 165 Revesz, 137, 164 Riemann integral, Lebesgue lemma, 307 -Stieltjes integral, Riesz representation theorem, 208 Robbins, 45,53, 135, 138, 160, 164, 209,221,286,312,375,384 Rogozin, 137 Rosenblatt, J., 353 Rosenblatt, M., 353 Rowwise independence, 138, 393, 454,459,460 Saks,29,112,209,312 Same type, 284 Sample (empirical) distribution, 284 space, 19 Samuel, 143, 164 Samuels, 82 Sapogov, 312 Savage,238,269, Scheffe, 312

9 487 Schwarz inequality, 105 Second moment analogue ofwald's equation, 144, 253, 254 Section of a function, 185 of a set, 13 Semi-algebra, 20, 24 Set Borel,ll function, 19 measurable, 8 operation, 2 Shohat, 283, 303, 312 Siegmund, 164, 402 a-additive (countably additive), 19 a-algebra, 6 generated by, 7, 16 of permutable events, 232 a-finite, 18 measure, 165 partition, 18 Signed measure, 208 Simple function, 18 random walk, 98 Singular distribution, 270 jl-singular, 203 Slutsky, 272, 312 Snell,209 Space, 4, 8, 11, 19,95, 111, 168, 185 Spitzer, 135, 164,402 Stable distribution, Standard deviation, 106 normal,31 Stein, E., 402 Stirling, 53 formula, 45, 49 Stochastic larger, 183.<t'p-bounded stochastic sequence, 239 matrix, 238 sequence, 138,239 Stone, 102, 112, 147, 164 Stopping time, 138, 240 {X n }-time,138 Strassen, 403 Stratton, 83 Strong law of large numbers (SLLN),124 independent random variables, 42, , , interchangeable random variables, 235 martingale differences, 258, 415 pairwise independent random variables, 132 V-statistics, 263 arrays, 138, 393 Studden, 111 Subadditive set function, 19 Submartingale, 239 closed,240 convergence theorems, 246, 248, 406,407 Subtractive, 24 Success probability, 56 Supermartingale, 239 Support (spectrum), 28, 271 Symmetric Bernoulli distribution, 294 distribution, 72, 297 random variable, 72 Symmetrized random variable, 197 System A-,15 monotone, 15 Tail event, 64 function, 64 of a distribution, 49 a-algebra Tamarkin, 283, 303, 312 Taylor, 403 Tchebychev inequality, 40, 106

10 488 Teicher, 53, 112, 137, 163, 164, 268, 269,312,353,403,443 Three series theorem, 117, 123 Tight, 276 Titchmarsh, 302, 303, 306, 312 Total variation, 208 Triangular distribution, 294 Truncation, 106, 110, 182,213 Vncorrelated, 106 Vniform distribution, 39, 293 Uniformly absolutely continuous, 208 bounded random variables, 116 integrable (u.i.) random variables, 93,94 integrable relative to distribution functions, 276, 277 V-statistics, 241, Central limit theorem, 326 degenerate, 259 decomposition of, 261 Strong law of large numbers, 263 Law of the iterated logarithm, 382 Van Beek, 323, 353 Variance, 105 Von Mises, 58, 83 Wald, 158, 162, 164 equation, 143, 144,253, 254,415 Weak compactness (sequential), 282 law of large numbers, 124, 128, 235, Weierstrass, 121 approximation theorem, 42 Weiss, I., 353 Widder, 178, 209 Wiener dominated ergodic theorem, 387,403 Wintner, 374, 378, 402 Wolfowitz, 347,402 Yadrenko, 149 Young, L. C, 112 Young, W. H., inequality, 184,424 Zero-one law, 97 Hewitt-Savage, 238 Kolmogorov,64 for interchangeable random variables, 238 see also Borel zero-one criterion Zolotarev, 353 Zygmund, 118, 121, 125, 164,386, 387,402,422,443

11 Springer Texts in Statistics (continued from page ii) Peters: Counting for Something: Statistical Principles and Personalities Pfeiffer: Probability for Applications Pitman: Probability Robert: The Bayesian Choice: A Decision-Theoretic Motivation Santner and Duffy: The Statistical Analysis of Discrete Data Saville and Wood: Statistical Methods: The Geometric Approach Sen and Srivastava: Regression Analysis: Theory, Methods, and Applications Whittle: Probability via Expectation, Third Edition Zacks: Introduction to Reliability Analysis: Probability Models and Statistical Methods