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Shiryaev A.N. — Probability
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Название: ProbabilityАвтор: Shiryaev A.N. Аннотация: This book contains a systematic treatment of probability from the ground up, starting with intuitive ideas and gradually developing more sophisticated subjects, such as random walks, martingales, Markov chains, ergodic theory, weak convergence of probability measures, stationary stochastic processes, and the Kalman-Bucy filter. Many examples are discussed in detail, and there are a large number of exercises. The book is accessible to advanced undergraduates and can be used as a text for self-study. This new edition contains substantial revisions and updated references. The reader will find a deeper study of topics such as the distance between probability measures, metrization of weak convergence, and contiguity of probability measures. Proofs for a number of some important results which were merely stated in the first edition have been added. The author included new material on the probability of large deviations, and on the central limit theorem for sums of dependent random variables.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Издание: Second EditionГод издания: 1995Количество страниц: 644Добавлена в каталог: 11.04.2008Операции: Положить на полку |
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Chap.VI 415 338 76 170 225 134 A posteriori probability 27 a priori probability 27 a.e. 185 a.s. 185 Absolute continuity with respect to P 195 524 Absolute moment 182 Absolutely continuous, distributions 155 Absolutely continuous, functions 156 Absolutely continuous, measures 155 Absolutely continuous, probability measures 195 524 Absolutely continuous, random variables 171 Absorbing 113 588 Accessible state 569 Algebra induced by a decomposition 12 Algebra of events 12 128 Algebra of sets 132 139 Algebra, 133 139 Algebra, correspondence with decomposition 82 Algebra, smallest 140 Algebra, tail 380 Almost everywhere 185 Almost invariant 407 Almost periodic 417 Almost surely 185 Amplitude 418 Aperiodic state 572 Appropriate sets, principle of 141 Arcsine law 102 Arithmetic properties 569 Asymptotic algebra 380 Asymptotic properties 573 Asymptotically, infinitesimal 337 Asymptotically, unbiased 443 Asymptotics 536 atoms 12 Attraction of characteristic functions 298 Autoregression, autoregressive 419 421 Average time of return 574 Averages, moving 419 421 437 Axioms 138 Backward equation 117 Balance equation 420 Ballot Theorem 107 Banach space 261 Barriers 588 590 Bartlett’s estimator 444 Basis, orthonormal 267 Bayes’s formula 26 Bayes’s theorem 27 230 Bernoulli, distribution 155 Bernoulli, James 2 Bernoulli, law of large numbers 49 Bernoulli, random variable 34 46 Bernoulli, scheme 30 45 55 70 Bernstein, inequality 55 Bernstein, polynomials 54 Bernstein, proof of Weierstrass’ theorem 54 Bernstein, S.N. 4 307 Berry — Esseen theorem 63 374 Bessel’s inequality 264 Best estimator 42 69 see Beta distribution 156 Bilateral exponential distribution 156 Binary expansion 131 394 Binomial distribution 17 155 Binomial distribution, negative 155 Binomial random variable 34 Birkhoff, G.D. 404 Birthday problem 15 Bochner — Khinchin theorem viii 287 409 Borel — Cantelli lemma 255 Borel — Cantelli — Levy theorem 518 Borel, algebra 139 Borel, E. 4 Borel, function 170 Borel, rectangle 145 Borel, sets 143 147 Borel, space 229 Borel, zero-or-one law 380 Bose — Einstein 10 Boundary 536 Bounded variation 207 Branching process 115 Brownian motion 306 Buffon’s needle 224 Bunyakovskii, V.Ya. 38 192 Burkholder’s inequality 499 Canonical decomposition 544 Canonical probability space 247 Cantelli, F.P. 255 388 Cantor, G., diagonal process 319 Cantor, G., function 157 158 Caratheodory’s theorem 152 Carleman’s test 296 Cauchy Bunyakovskii inequality 38 192 Cauchy criterion for almost sure convergence 258 Cauchy criterion for convergence in mean-p 260 Cauchy criterion for convergence in probability 259 Cauchy, distribution 156 344 Cauchy, inequality 38 Cauchy, sequence 253 Central limit theorem 4 322 326 348 Central limit theorem for dependent variables 541 Central limit theorem, nonclassical condition for 337 Certain event 11 136 Cesaro limit 582 Change of variable in integral 196 Chapman, D.G. 116 248 566 Characteristic function of distribution 4 274 Characteristic function of random vector 275 Characteristic function of set 33 Charlier, C.V.L. 269 Chebyshev inequality 47 55 192 Chebyshev, P.L. 3 321 Chi, chi-squared distributions 156 243 Class 515 Class convergence-determining 315 Class determining 315 Class monotonic 140 Class of states 570 Classical method 15 Classical models 17 Classification of states 569 573 Closed linear manifold 267 coin tossing 1 5 17 33 83 131 Coincidence problem 15 Collectively independent 36 Communicating states 570 Compact relatively 317 Compact sequentially 319 Compensator 482 Complement 11 136 Complete function space 260 Complete probability measure 154 Complete probability space 154 Completely nondeterministic 447 Conditional distribution 227 Conditional expectation in the wide sense 264 274 Conditional expectation with respect to 214 Conditional expectation with respect to decomposition 78 Conditional expectation with respect to event 220 Conditional expectation with respect to set of variables 81 Conditional probability 23 77 221 Conditional probability variance 83 214 Conditional probability with respect to a 212 214 Conditional probability with respect to a decomposition 77 212 Conditional probability with respect to a random variable 77 214 Conditional probability, regular 227 Conditional probability, Wiener process 307 Conditionally Gaussian 466 Confidence interval 74 Consistency Property 163 246 Consistent estimator 71 521 535 Construction of a process 245 246 Continuity theorem 322 Continuous at 153 164 Continuous from above or below 134 Continuous time 177 Convergence of martingales and submartingales 508 515 Convergence of probability measures Chap.III 308 Convergence of random variables: equivalences 252 Convergence of sequences see Convergence of sequences Convergence of sequences at points of continuity 253 Convergence of sequences in distribution 252 325 353 Convergence of sequences in general 310 Convergence of sequences in mean 252 Convergence of sequences in mean of order p 252 Convergence of sequences in mean square 252 Convergence of sequences in measure 252 Convergence of sequences in probability 252 348 Convergence of sequences with probability 252 Convergence of sequences, almost everywhere 252 353 Convergence of sequences, almost sure 252 353 Convergence of sequences, dominated 187 Convergence of sequences, monotone 186 Convergence of sequences, weak 309 311 Convergence of series 384 Convergence-determining class 315 Convolution 241 377 Coordinate method 247 Correlation, coefficient 41 234 Correlation, function 416 Correlation, maximal 244 Counting measure 233 Covariance 41 232 293 Covariance, function 306 416 Covariance, matrix 235 Cramer condition 400 Cramer transform 401 Cramer — Lundberg model 559 Cramer — Wold method 549 Cumulant 290 Curve of regression 238 Curvilinear boundary 536 Cyclic property 571 Cyclic subclasses 571 Cylinder set 146 d-system 142 Davis’s inequality 499 De Moivre — Laplace limit theorem 62 de Moivre, A. 2 49 Decomposition of martingale 507 Decomposition of probability measure 525 Decomposition of random sequence 447 Decomposition of set 12 292 Decomposition of submartingale 482 Decomposition, canonical 544 Decomposition, countable 140 Decomposition, Doob 482 Decomposition, Krickeberg 507 Decomposition, Lebesgue 525 Decomposition, of 12 140 Decomposition, trivial 80 Degenerate, distribution 298 Degenerate, distribution function 298 Degenerate, random variable 298 Delta function 298 Delta, Kronecker 268 Density of distribution 156 Density of measure with respect to a measure 196 Density of random variable 171 Density, Gaussian 66 156 161 238 Density, n-dimensional 161 Dependent random variables 103 Dependent random variables, central limit theorem for 541 Derivative, Radon — Nikodym 196 Detection of signal 462 Determining class 315 Deterministic 447 Deterministic, regularity 1 Dichotomy, Hajek — Feldman 533 Dichotomy, Kakutani 529 Difference of sets 11 136 Direct product 31 144 151 Dirichlet’s function 211 Discrete measure 155 Discrete random variable 171 Discrete time 177 Discrete uniform density 155 Discrete version of Ito’s formula 554 Disjoint 136 dispersion 41 Distance in variation 355 376 distribution see also Distribution function Probability Distribution function 34 35 152 171 Distribution function of functions of random variables 36 239ff. Distribution function of sum 36 241 Distribution function, absolutely continuous 156 Distribution function, degenerate 288 Distribution function, discrete 155 Distribution function, finite-dimensional 246 Distribution function, generalized 158 Distribution function, n-dimensional 160 Distribution of objects in cells 8 Distribution of process 178 Distribution of sum 36 Distribution with density (list) 156 Distribution, Bernoulli 155 Distribution, Beta 156 Distribution, binomial 17 18 155 Distribution, Cauchy 156 344 Distribution, chi, chi-squared 243 Distribution, conditional 212 Distribution, discrete (list) 155 Distribution, discrete uniform 155 Distribution, entropy of 51 Distribution, ergodic 118 Distribution, exponential 156 Distribution, gamma 156 Distribution, Gaussian 66 156 161 293 Distribution, geometric 155 Distribution, hypergeometric 21 Distribution, infinitely divisible 341 Distribution, initial 112 565 Distribution, invariant 120 Distribution, limit 545 Distribution, lognormal 240 Distribution, multidimensional 160 Distribution, multinomial 20 Distribution, negative binomial 155 Distribution, normal 66 156 Distribution, Poisson 64 155 Distribution, polynomial 20 Distribution, probability 33 Distribution, stable 341 Distribution, stationary 120 Distribution, Student’s 155 244 Distribution, t- 155 244 Distribution, two-sided exponential 155 Distribution, uniform 156 Dominated convergence 187 Dominated sequence 496 Doob, J.L. 482 485 492 Doubling stakes 89 481
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