|
|
 |
| Авторизация |
|
|
|
| Поиск по указателям |
|
|
|
|
|
|
|
 |
|
|
|
| Doob J.L. — Stochastic processes |
|
|
|
| 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  501
Rational spectral densities: in  (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,  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
|
|
|
| Реклама |
|
|
|
|
|
О проекте