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Doob J.L. — Stochastic processes
Doob J.L. — Stochastic processes

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Название: Stochastic processes

Автор: Doob J.L.

Аннотация:

The theory of stochastic processes has developed so much in the last twenty years that the need for a systematic account of the subject has been felt, particularly by students and instructors of probability. This book fills that need. While even elementary definitions and theorems are stated in detail, this is not recommended as a first text in probability and there has been no compromise with the mathematics of probability. Since readers complained that omission of certain mathematical detail increased the obscurity of the subject, the text contains various mathematical points that might otherwise seem extraneous. A supplement includes a treatment of the various aspects of measure theory. A chapter on the specialized problem of prediction theory has also been included and references to the literature and historical remarks have been collected in the Appendix.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: Wiley classics library edition

Год издания: 1990

Количество страниц: 654

Добавлена в каталог: 04.12.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Absolute centering constants      110
Absolute probabilities of a Markov process      172 191 214
Absolutely continuous set function      611
Absolutely continuous spectral distribution      498 532
Absorbing barrier      243
Additive process      391
Adjunction, extension of a process by      71
Admissible Borel field      208
Backward equations: chain case      254 272
Backward equations: diffusion case      274
Backward equations: general purely discontinuous case      270 273
Baire functions      600
Baire functions, generalized      613
Bessel's inequality      152
Borel field      599
Borel measurable function      600
Borel set      600
Borel set, generalized      613
Borel — Cantelli lemma      104
Brownian movement process: condition that a process with independent increments be one      420
Brownian movement process: conditions that a martingale be one      384
Brownian movement process: definition      97
Brownian movement process: general discussion      392
Campbell's theorem      433
Card mixing      186
Centering constants      110
Centering constants, absolute      110
Centering function of a process with independent increments      407
Central limit theorem: Markov processes      221
Central limit theorem: martingales      383
Central limit theorem: sums of mutually independent random variables      137
Chapman — Kolmogorov equation      88 235 255
Characteristic function of a distribution      37
Characteristic function, application to central limit theorem      139
Characteristic function, application to convergence of series of mutually independent random variables      115
Closed linear manifold      75 149
Complete measure      5 606 623
Completely additive set function      604
Conditional probabilities and expectations, conditional probability distribution      26 623
Conditional probabilities and expectations, conditional probability distribution, wide sense      29
Conditional probabilities and expectations, definition      15
Conditional probabilities and expectations, Gaussian case      76
Conditional probabilities and expectations, iterated      35
Conditional probabilities and expectations, wide sense      155
Consequent: integer of a Markov chain      176
Consequent: set of a Markov process      206
Consistency of an estimation procedure      633
Continuity properties of stochastic process sample functions: Brownian movement      393
Continuity properties of stochastic process sample functions: Markov chain      246 248 265
Continuity properties of stochastic process sample functions: Markov process      258 260 266 267 388
Continuity properties of stochastic process sample functions: martingale      361
Continuity properties of stochastic process sample functions: process with independent increments      388 420 422
Convergence: in distribution      9
Convergence: stochastic, in measure, in mean, with probability one      8
Convex function of a semi-martingale or martingale      295
Convolution      78
Covariance function: characterization in stationary case      473 518
Covariance function: definition in stationary case      95
Covariance function: definition in stationary case, multidimensional case      596
Covariance function: general characterization      72
Cyclically moving sets: general state space      211
Cyclically moving sets: Markov chain      177
Cylinder set      600
D: Hypothesis      192
Density of a distribution      6
Derivative of a set function relative to a net      343 612
Determined: set determined by conditions on specified random variables      292
Deterministic process      564
Difference field      511
Difference manifold      513
Difference sets      511
Differential process      391
Differentiation of sample functions: process with stationary increments      558
Differentiation of sample functions: stationary process      535
Diffusion equations      275
Diffusion-type process      273
Distribution function      6
Distribution function, density      6
Distribution function, multivariate      6
Dominated: process dominated by a semi-martingale      297
Ergodic classes of a Markov process: chain case      179
Ergodic classes of a Markov process: continuous state space      209 210
Ergodic theorem: continuous parameter      515
Ergodic theorem: discrete parameter      464
Estimation of covariance and spectral distribution functions: continuous parameter      531
Estimation of covariance and spectral distribution functions: discrete parameter      493
Expectation of a random variable      8
Extension of a stochastic process by adjunction      71
Fair game      299
Favorable game      299
Field of sets      599
Filter      638
Fixed point of discontinuity of a stochastic process      357
Fokker — Planck equation      275
Forward equations: chain case      254 272
Forward equations: diffusion case      274
Forward equations: general purely discontinuous case      270 273
Fourier series      150
Fourier transform of a process with orthogonal increments      434
Function space type, process of      67
Fundamental theorem of sequential analysis      352
Gain of a linear operation on a stationary process      534
Game of chance: fair, favorable game      299
Game of chance: invariance of fairness and favorableness under optional sampling      302 373 376
Game of chance: invariance of fairness and favorableness under optional skipping      309
Game of chance: invariance of fairness and favorableness under optional stopping      300
Game of chance: system      145
Gaussian process: conditional expectations in one      390
Gaussian process: conditions that a process with independent increments be one      420
Gaussian process: criterion for existence      72
Gaussian process: definition      71
Gaussian process: metric transitivity of      637
Harmonic analysis of a stationary process      469 517
Independent increments, process with      see Chapter VIII
Independent increments, process with: definition      96
Independent increments, process with: sample function continuity of      388 422
Independent increments, process with: stationary increments      97 512
Independent random variables      see Chapter III
Independent random variables: definition      7
Independent random variables: processes with      78 102
Infinitely divisible distribution      128
Integral with independent random elements      391
Integration in infinitely many dimensions      342
Integration of sample functions      62
Integration of sample functions, in a stationary process      538
Invariant random variables: of a stationary Markov process      460
Invariant random variables: under isometric transformations (wide sense)      463
Invariant random variables: under measure-preserving transformations (strict sense)      457 610
Invariant set: minimal invariant set of a Markov process      206
Invariant set: of a Markov process      206 460
Invariant set: of measure-preserving transformations      457 510
Isometric transformations      461
Isometric transformations, semi-group and group of      512
Jensen's inequality for conditional probabilities      33
Jump      246
Large number's, law of: definition      122
Large number's, law of: for Markov processes      218
Large number's, law of: for processes with stationary independent increments      364
Large number's, law of: for strictly stationary processes      95 465 515
Large number's, law of: for sums of independent random variables      123
Large number's, law of: for sums of independent random variables, with a common distribution      142 341
Large number's, law of: for sums of orthogonal random variables      158
Large number's, law of: for wide sense stationary processes      489 529
Least squares approximation      76
Least squares approximation, linear      77
Lebesgue — Stieltjes measure      607
Likelihood ratio      93 348
Linear manifold      149
Linear manifold, closed      75
Linear operations on stationary processes: continuous parameter      534
Linear operations on stationary processes: discrete parameter      500
Lower semi-martingale      294
Markov chain: application to card mixing      186
Markov chain: continuous parameter      235 265 271 388
Markov chain: discrete parameter      170
Markov process      see Chapters V and VI (see also “Markov chain” “Stationary
Markov process: covariance function in wide sense case      233
Markov process: definition      80
Markov process: definition, wide sense      90
Markov property      81
Markov transition function      255
Markov transition matrix function      236
Martingale      see Chapter VII
Martingale: defined by stochastic integrals      444
Martingale: definition      91
Martingale: relative to specified Borel fields      294
Martingale: wide sense      164
Measurability: of a stochastic process      60
Measurability: of sample functions      62 60
Measurable set on the sample space of specified random variables      19
Measure function      605
Measure, complete      5 606
Measure, Lebesgue — Stieltjes      607
Measure, probability      605
Measure-preserving point transformations      452 617
Measure-preserving point transformations, translation semi-group, group of      507
Measure-preserving set transformations      452
Measure-preserving set transformations, translation semi-group, group of      507
Metrically transitive Markov process      460
Metrically transitive process relative to the difference field      511
Metrically transitive process with independent random variables      460
Metrically transitive process with orthogonal random variables      464
Metrically transitive process with stationary (wide sense) orthogonal increments      514
Metrically transitive process with stationary increments      512
Metrically transitive stochastic process      457
Metrically transitive stochastic process, wide sense      463
Metrically transitive transformation      457
Metrically transitive transformation, wide sense      463
Metrically transitive translations of [0, 1] modulo one      508
Minimal invariant set of a Markov process      206
Molecular distributions      404
Moving averages, process of: continuous parameter      532
Moving averages, process of: discrete parameter      498
Moving averages, process of: finite average      504
Moving point of discontinuity      357
Multidimensional prediction      594
Multiple Markov process      89
Multiple Markov process, application to card mixing      186
Optional sampling: continuous parameter      366
Optional sampling: discrete parameter      301
Optional skipping      310
Optional stopping: continuous parameter      366
Optional stopping: discrete parameter      300
Orthogonal increments, process with      see Chapter IX
Orthogonal increments, process with: definition      99
Orthogonal increments, process with: metric transitivity of      514
Orthogonal random variables, processes with      see Chapter IV
Orthogonal random variables, processes with: definition      79
Orthogonality      74
Orthogonalization      151
Poisson process: application to molecular and stellar distributions      404
Poisson process: definition      98
Poisson process: general discussion      398
Polynomial approximation      562
Positive definite function: continuous argument      519
Positive definite function: discrete argument      473
Prediction      see Chapter XII
Prediction: by way of a stochastic differential equation      550
Prediction: multiple Markov discrete parameter      506
Probability measure      605
Projection: definition      155
Projection: wide sense martingale limit theorems      166
Purely random events      400
q-bounded set      260 265
Random events      400
Random variable: definition      5
Random variable: on the sample space of specified random variables      19
Random walk      308
Rational spectral densities: (discrete parameter) in $e^{2\pi i\lambda}$      501
Rational spectral densities: in $\lambda$ (continuous parameter)      542
Reduction procedure      204
Reflection principle      393
Regular stochastic process      564
Representation of a family of random variables      12
Representation of a family of random variables, applied to conditional expectations      33
Representation of a family of random variables, detailed justification      623
Sample functions: definition      11
Sample functions: differentiation      535 558
Sample functions: integration      62
Sample functions: measurability      22
Sample space: definition      3
Sample space: function or set measurable on the sample space of specified random variables      19
Semi-martingale      see Chapter VII
Semi-martingale: definition      292
Semi-martingale: relative to specified Borel fields      294
Separability of a stochastic process      52
Separability, relative to a specified class of sets      51
Sequential analysis, application of martingale theory to: continuous parameter      380
Sequential analysis, application of martingale theory to: discrete parameter      350
Series: Fourier series      150
Series: of mutually independent random variables      105 335
Series: of orthogonal random variables      155
Series: of power series type      159
Series: three series theorem      111
Set of increase of a singular set function      611
Shift transformation: continuous parameter, isometric case      512
Shift transformation: continuous parameter, measure-preserving case      510
Shift transformation: discrete parameter, isometric case      462
Shift transformation: discrete parameter, measure-preserving case      455
Singular component of a set function      611
Singular set function      611
Singular set of a singular set function      611
Smoluchovski equation      88
Spectral decomposition of a stationary process: continuous parameter      529
Spectral decomposition of a stationary process: discrete parameter      486
Spectral density of a stationary process: continuous parameter      522
Spectral density of a stationary process: discrete parameter      476
Spectral distribution function of a stationary process: continuous parameter      522
Spectral distribution function of a stationary process: discrete parameter      476
Spectral representation of a stationary process: continuous parameter      527
Spectral representation of a stationary process: discrete parameter      481
Spectrum of a stationary process      476
Standard extension of a stochastic process      69
Standard modification of a stochastic process      66
Standard pair of q-functions      265
Stationary (wide sense) increments, process with      99 551
Stationary independent increments      364
Stationary Markov process (wide sense)      437 506 523 550 566
Stationary Markov process, Gaussian case      218 234 506
Stationary Markov transition function      256
Stationary process      see Chapters X XI
Stationary process: definition (multidimensional wide sense)      596
Stationary process: definition (strict sense)      94
Stationary process: definition (wide sense)      95
Stellar distributions      404
Step function      426
Step function, $(t, \omega)$ step function      438
Stochastic difference equations      503
Stochastic differential equations: diffusion type      273
Stochastic differential equations: satisfied by a stationary process      546 559
Stochastic integral      62 426 436 540
Stochastic matrix      172
Stochastic process (see individual types under their own names): definition      46
Stochastic transition function      190
Stochastic transition function, density      193
Stochastically definite process      625
Strict sense concepts      77
Temporally homogeneous process      96
Three series theorem      111
Transient set of a Markov process      210
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