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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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Doubly stochastic 587 Duration of random walk 90 Dvoretzky’s inequality 508 Efficient estimator 71 Eigenvalue 130 Electric circuit 32 Elementary events 5 136 Elementary probability theory Chap.I Elementary stochastic measure 424 Empty set 11 136 entropy 51 Equivalent measures 524 Ergodic sequence 407 Ergodic theorems 110 409 413 Ergodic theorems, maximal 410 Ergodic theorems, mean-square 438 Ergodic theory 409 Ergodic transformation 408 Ergodicity 118 409 581 Errors of observation 2 3 Errors, laws of 298 Errors, mean-square 43 Esseen’s inequality 296 Essential state 569 Essential supremum 261 Estimation 70 237 440 454 Estimator 42 70 237 Estimator for parameters 472 520 535 Estimator of spectral quantities 442 Estimator, Bartlett’s 444 Estimator, best 42 69 Estimator, consistent 71 Estimator, efficient 71 Estimator, linear 43 Estimator, optimal 70 237 303 454 461 463 469 Estimator, Parzen’s 445 Estimator, unbiased 71 440 Estimator, Zhurbenko’s 445 Events 5 10 136 Events, certain 11 136 Events, elementary 5 Events, impossible 11 136 Events, independent 28 29 Events, mutually exclusive 136 Events, symmetric 382 Existence of limits and stationary distributions 582ff. Expectation 37 182 Expectation of function 55 Expectation of maximum 45 Expectation of random variable with respect to decomposition 76 Expectation of random variable with respect to decomposition, set of random variables 81 Expectation of sum 38 Expectation, inequalities for 192 193 Expectationof random variable with respect to decomposition, 212 Expected value 37 Exponential distribution 156 Exponential random variable 156 244 245 Extended random variable 178 Extension of a measure 150 163 249 427 Extrapolation 453 F-distribution 156 Fair game 480 Fatou’s Lemma 187 211 Favorable game 89 480 Feldman, J. 533 Feller, W. 597 Fermat, P.de 2 Fermi—Dirac 10 Filter 434 464 Filter, physically realizable 451 Filtering 453 464 Finer decomposition 80 Finite second moment 262 Finite-dimensional distribution function 246 Finitely additive 132 424 First arrival 129 574 First exist 123 First return 94 129 574 Fisher’s information 72 Forward equation 117 Foundations Chap.II 131 Fourier transform 276 frequencies 418 frequency 46 Frequency characteristic 434 Fubini’s Theorem 198 Fundamental inequalities (for martingales) 492 Fundamental sequence 253 258 Gamma distribution 156 343 Garcia, A. viii 410 Gauss Markov process 307 Gauss, C. F. 3 Gaussian density 66 156 161 236 Gaussian distribution 66 156 161 293 Gaussian measure 268 Gaussian random variables 234 243 298 Gaussian random vector 299 Gaussian sequence 306 413 439 441 466 Gaussian systems 297 305 Generalized Bayes theorem 231 Generalized distribution function 158 Generalized martingale 476 Generalized submartingale 476 523 Geometric distribution 155 Gnedenko, B.V. vii 510 542 Gram determinant 265 Gram — Schmidt process 266 Haar functions 271 482 Hajek — Feldman dichotomy 533 Hardy class 452 Harmonics 418 Hartman, P. 372 Hellinger integral 3 Helly’s theorem 319 Herglotz, G. 421 Hermite polynomials 268 Hewitt, E. 382 Hilbert space 262 Hilbert space, complex 275 416 Hilbert space, separable 267 Hilbert space, unitary 416 Hinchin see Khinchin history 597—602 Hoelder inequality 193 Huygens, C. 2 Hydrology 420 421 Hypergeometric distribution 21 Hypotheses 27 I.o. 137 Impossible event 11 136 Impulse response 434 Increasing sequence 137 Increments independent 306 Increments uncorrelated 109 306 Indecomposable 580 Independence 27 Independence, linear 265 286 Independent algebras 28 29 Independent events 28 29 Independent functions 179 Independent increments 306 Independent random variables 36 77 81 179 380 513 Indicator 33 43 Inequalities, Bernstein 55 Inequalities, Berry — Esseen 333 Inequalities, Bessel 264 Inequalities, Burkholder 499 Inequalities, Cauchy — Bunyakovskii 38 192 Inequalities, Cauchy — Schwarz 38 Inequalities, Chebyshev 3 321 Inequalities, Davis 499 Inequalities, Dvoretzky 508 Inequalities, Hoelder 193 Inequalities, Jensen 192 233 Inequalities, Khinchin 347 498 Inequalities, Kolmogorov 496 Inequalities, Levy 400 Inequalities, Lyapunov 193 Inequalities, Marcinkiewicz — Zygmund 498 Inequalities, Markov 598 Inequalities, martingale 492 Inequalities, Minkowski 194 Inequalities, nonuniform 376 Inequalities, Ottaviani 507 Inequalities, Rao — Cramer 73 Inequalities, Schwarz 38 Inequalities, two-dimensional Chebyshev 55 Inessential state 569 Infinitely divisible 341 Infinitely many outcomes 131 information 72 Initial distribution 112 565 Innovation sequence 448 Insurance 558 Integral equation 208 Integral Lebesgue 180 Integral Lebesgue — Stieltjes 183 Integral Riemann 183 205 Integral Riemann — Stieltjes 205 Integral stochastic 423 Integral theorem 62 Integration by parts 206 Integration by substitution 211 Intensity 418 Interpolation 453 Intersection 11 136 Introducing probability measures 151 Invariant set 407 413 Inversion formulas 283 295 Ionescu Tulcea, C. T. vii 249 Ising model 23 Isometry 430 Iterated logarithm 395 Ito’s formula for Brownian motion 558 Jensen’s Inequality 192 233 Kakutani dichotomy 527 528 Kakutani — Hellinger distance 363 Kalman — Bucy filter 464 Khinchin, A.Ya 287 468 Kolmogorov — Chapman equation 116 248 566 Kolmogorov, A.N. vii 3 4 384 395 498 542 Kolmogorov, axioms 131 Kolmogorov, inequality 384 Kolmogorov’s theorems, convergence of series 384 Kolmogorov’s theorems, existence of process 246 Kolmogorov’s theorems, extension of measures 167 Kolmogorov’s theorems, iterated logarithm 395 Kolmogorov’s theorems, stationary sequences 453 455 Kolmogorov’s theorems, strong law of large numbers 366 389 391 Kolmogorov’s theorems, three-series theorem 387 Kolmogorov’s theorems, two-series theorem 386 Kolmogorov’s theorems, zero-or-one law 381 Krickeberg’s decomposition 507 Kronecker, delta 268 Kronecker, L. 390 Kullback information 368 Laplace, P.S. 2 55 Large deviation 69 402 Law of Large Numbers 45 49 325 Law of large numbers for Markov chains 122 Law of large numbers for square-integrable martingales 519 Law of large numbers, Poisson’s 599 Law of Large Numbers, strong 388 Law of the iterated logarithm 395 Least squares 3 Lebesgue — Stieltjes integral 197 Lebesgue — Stieltjes measure 158 205 Lebesgue, H., change of variable in 196 Lebesgue, H., decomposition 366 525 Lebesgue, H., derivative 366 Lebesgue, H., dominated convergence theorem 187 Lebesgue, H., integral 180 181 Lebesgue, H., measure 154 159 LeCam, L. 377 Levy — Khinchin representation 347 Levy — Khinchin theorem 344 Levy — Prokhorov metric 349 Levy, P., convergence theorem 510 Levy, P., distance 316 Levy, P., inequality 400 Likelihood ratio 110 lim inf, lim sup 137 Limit theorems 55 Limits under, expectation signs 180 Limits under, integral signs 180 Lindeberg condition 328 Lindeberg — Feller theorem 334 Linear manifold 264 Linearly independent 265 Liouville’s theorem 406 Lipschitz condition 512 Local limit theorem 55 56 Local martingale, submartingale 477 Locally absolutely continuous 524 Locally bounded variation 206 lognormal 240 Lottery 15 22 LOWER function 396 Lyapunov, A.M. 3 322 Lyapunov, condition 332 Lyapunov, inequality 193 Macmillan’s theorem 59 Marcinkiewicz — Zygmund inequality 498 Marcinkiewicz’s theorem 288 Markov chains 110 Chap.VIII 251 564 Markov chains, classification of 569 573 Markov chains, discrete 565 Markov chains, examples 113 587 Markov chains, finite 565 Markov chains, homogeneous 113 565 Markov, A.A. viii 3 321 Markov, depencence 564 Markov, process 248 Markov, property 112 127 564 Markov, time 476 Martingale 103 Chap.VII 474 Martingale transform 478 Martingale, convergence of 508 Martingale, generalized 476 Martingale, in gambling 480 Martingale, inequalities for 492 Martingale, local 477 Martingale, oscillations of 503 Martingale, reversed 484 Martingale, sets of convergence for 515 Martingale, square-integrable 482 493 538 Martingale, uniformly integrable 512 Martingale-difference 481 543 559 Mathematical expectation 37 76 see Mathematical foundations Chap.II 131 Matrix covariance 235 Matrix of transition probabilities 112 Matrix transition 112 Matrix, doubly stochastic 587 Matrix, orthogonal 235 265 Matrix, stochastic 113 Maximal correlation coefficient 244 Maximal ergodic theorem 410 Maxwell — Boltzmann 10 Mean duration 90 489
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