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| Àâòîðèçàöèÿ |
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| Ïîèñê ïî óêàçàòåëÿì |
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| Borovkov A.A. — Ergodicity and Stability of Stochastic Processes |
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| Ïðåäìåòíûé óêàçàòåëü |
ALOHA (algorithm) 167 502
ALOHA applications 538
Aperiodicity condition, Markov chains 137
Approaching times method 45—49 317—393
Arzela — Ascoli theorem 72
Asmussen processes 236
Autoregression generalized 91—97
Autoregression linear 95
Axes attracting 345 357
Axes ergodic 345 349
Axes ergodic N-neighborhood 351
Axes repelling 345 346 349
Axes transitory 349 358
Banach space 94 108
Bennett — Hoeffding inequality 486 487 488 489
Bernoulli sequence 377
Bochner integral 108
Borel functions 397
Borel s-algebra 4 108 114 138 140 160 165 166 173 194 230 282
Borel sets 175 177 417 535
Borel subsets 245
Boundary functionals, distributions, asymptotics 486—493
Boundedness, in probabilities 137 209—215
Buffered random access systems, use of term 539
Canonical factorization 301
Central limit theorems 91
Chebyshev inequality 23 213 306 481
Communication channels, queueing systems 538—544
Communication networks see also Jackson networks; queueing networks
Communication networks ergodicity 465—558
Communication networks stability 465—558
Communication networks, recursive chains in 166—167
Communication systems see also polling systems; queueing systems; random access communication systems
Communication systems stochastic processes 501
Communication systems, types of 554
Cones, three-dimensional, random walks 262
Convergence rates estimates 101
Convergence rates Jackson networks 536—537
Convergence rates polling systems 521—522
Convergence rates random access communication systems 547
Convergence rates stochastically recursive sequences 162
Convergence, majorized 52
Coupling method, in theorems 25—27
Coupling-convergence see also strong coupling-convergence
Coupling-convergence stochastically recursive sequences 148—149 153—154
Cramer transform 91
Cramer’s condition 301 325 326 350 474 484 485 488
Cramer’s violation 300
Cycles envelopes 379
Cycles tags 34
Cycles truncated 34
Density, diffusion processes 399
Deviations, large, oscillating random walks 292
Deviations, large, probabilities of 293—316 467—500 521—522
Deviations, large, random walks 285—286
Differential equations, stochastic 223 396 411
Differential equations, transition function as solution 398
Diffusion coefficients 398
Diffusion processes density 399
Diffusion processes ergodicity 395—411
Diffusion processes ergodicity conditions 400—411
Diffusion processes Feller property 414
Diffusion processes stability theorems 414
Diffusion processes stationary distributions 410—411
Diffusion processes with reflection 397—398
Diffusion processes, asymptotically spatially homogeneous 407—410
Diffusion processes, multi-dimensional 416
Diffusion processes, multi-dimensional, ergodicity and stability 395—416
Diffusion processes, one-dimensional 409 410 414
Diffusion processes, two-dimensional 402 409
Diffusion processes, unbounded 399 404
Dirac d-function 399
Distributions see also Gamma-distribution; Poisson distributions; stationary distributions
Distributions Kantorovich — Wasserstein distance 56
Distributions Levy — Prokhorov distance 85 101 131 230
Distributions probability, invariant 17
Distributions rate of convergence 90—91
Distributions stationary, stability 49—53
Distributions weak convergence 60
Distributions, distance between 131
Doob inequality 477
Drift coefficients 398
Edges, definition 278
Embedded processes, applications 236
Embedded sequences, definition 230
Environments, random, Markov chains in 135 165—178
Equilibrium identities, Markov chains 106—114
Ergodicity and transition probabilities 67—80
Ergodicity communication networks 465—558
Ergodicity conditions 103
Ergodicity conditions and random transformations 78—108
Ergodicity conditions for Lipschitz transformations 97—102 103
Ergodicity conditions for monotonic transformations 79—82
Ergodicity conditions Harris-type 524
Ergodicity conditions necessity 518—521
Ergodicity conditions sufficient 184—195 501
Ergodicity criteria, in arbitrary dimensions 272—274
Ergodicity diffusion processes 395—410
Ergodicity diffusion processes conditions 400—411
Ergodicity diffusion processes, multi-dimensional 395—416
Ergodicity for Lipschitz contracting transformations 82—91
Ergodicity general theorems 1—241
Ergodicity in the mean 55—67
Ergodicity in the mean Ioshida theorem 67
Ergodicity in the mean, definition 60
Ergodicity Jackson networks, open 530—536
Ergodicity jump processes 411—413
Ergodicity main theorem 11—32
Ergodicity Markov chains conditions 11—14 118 136 137—138
Ergodicity Markov chains early studies 3
Ergodicity Markov chains on half-lines 284—286
Ergodicity Markov chains proofs 101—107
Ergodicity Markov chains, Harris-type conditions 135—162
Ergodicity Markov chains, multi-dimensional 243—463
Ergodicity Markov chains, one-dimensional 277—316
Ergodicity Markov chains, two-dimensional 257—272 317—368
Ergodicity Markov processes 243—463
Ergodicity Markov processes continuous-time 225—230 395
Ergodicity of stochastic processes, in continuous and discrete time 221—241
Ergodicity oscillating random walks 72
Ergodicity partial 504
Ergodicity polling systems, of first kind 506—521 549—543
Ergodicity proofs, problems 246
Ergodicity queueing networks 501—558
Ergodicity random walks conditions 369—372
Ergodicity random walks, asymptotically homogeneous 317—343
Ergodicity random walks, multi-dimensional 281—282 503
Ergodicity recursive chains 178—195
Ergodicity stochastically recursive sequences, Harris-type conditions 135—162
Ergodicity theorems 56 374—375
Ergodicity theorems approaches 164—165
Ergodicity theorems convergence rates 14 19
Ergodicity theorems, simplified 29—30
Ergodicity weak convergence 76
Exponentiality conditions, relaxation 547—543
Exponentiality partial 502
Faces, definition 278
Factorization identities 294
Fatou lemma 475
Feller Markov chains stability 288 334
Feller Markov chains stability general theorems 114—118 293
Feller property 114
Feller property diffusion processes 414
FIFO services 504
Fokker — Planck equation 399
Foster — Mousstafa — Tweedie criterion 37
Functions see also Lipschitz continuous functions; Lyapunov functions; test functions; transition functions
Functions monotonicity 136 152
Functions, Borel 397
Functions, decreasing, notation 15
| Functions, non-increasing 15
Functions, Riemann integrable 232 235 258
Gamma-distribution random variables 452 456
Gamma-distribution, convergence to 420—421 425—434
Generalized autoregression 91—97
Half-lines, definition 278
Half-lines, infinitely distant 372
Half-lines, Markov chains on 284—289
Half-lines, positive, Markov chains on 417—425
Harris chains 550 (see also non-Harris chains)
Harris chains ergodicity 390—391
Harris chains ergodicity studies 55
Harris chains stability theorems 52—53 114
Harris chains, enlarged 31
Harris recurrence Markov chains 548
Harris recurrence theorems 5
Harris recurrence, concept of 3
Harris recurrence, definition 4
Hoelder inequality 479
Hopf’s theory 168
Hyperplanes, normal vectors 272
Hypersurfaces 252 274
Hypersurfaces tangent hyperplanes 274
Integrals convergence 69
Integrals, Bochner 108
Invariant measures, tightness conditions 118—130
Ioshida theorem 67
Jackson networks 529—537
Jackson networks convergence rates, estimates 536—537
Jackson networks ergodicity 411—413
Jackson networks regularity properties 412
Jackson networks stability 522
Jackson networks, closed 529
Jackson networks, open 529—531 548
Jackson networks, open, ergodicity 530—536
Jump processes, definitions 411
Kantorovich — Wasserstein distance between distributions 56
Kolmogorov equation 400
Kolmogorov theorem 187 194 224
Kolmogorov — Chapman equation 67 226 399
Kolmogorov, Andrei Nikolaevich (1903—1987), ergodicity studies 282
Ky-Fan distance 91
Lebesgue measure 70 82 160 195 228 233 283 287
Lebesgue measure, normalized 246
Lebesgue theorem 52 110 226 409
Levy — Prokhorov distance between distributions 85 101 131 230
Limit theorems, collective 424
Linear autoregression 95
Lipschitz condition 124 411 503
Lipschitz continuous functions 120
Lipschitz continuous functions, bounded 83
Lipschitz transformations contraction property 165
Lipschitz transformations, ergodicity conditions for 97—102 103
Lipschitz violation 125
Lyapunov conditions positivity 37—43
Lyapunov conditions, use of term 42
Lyapunov exponents, definition 95
Lyapunov functions 282
Lyapunov functions estimates 96
Lyapunov functions existence 252—256 267
Lyapunov functions in diffusion processes 400
Lyapunov functions methods 245—275
Lyapunov’s inequality 478
Majorants stationary 195—215 503
Majorants stationary construction 92—94
Markov chains see also Feller Markov chains; Harris chains; non-Harris chains; random walks; recursive chains
Markov chains and stochastically recursive ergodicity conditions 156—161
Markov chains and stochastically recursive sequences, compared 139—140
Markov chains aperiodicity condition 137
Markov chains arbitrary dimensions, criteria 272—274
Markov chains arbitrary, stability 130—133
Markov chains characterization 178
Markov chains classification 278—279 284
Markov chains convergence rates, estimates 161
Markov chains curvilinear coordinates 249
Markov chains distributions 8—11
Markov chains distributions, finite-dimensional 156
Markov chains equilibrium identities 106—114
Markov chains ergodicity assumptions 6
Markov chains ergodicity conditions 11—14 118 136 137—138
Markov chains ergodicity early studies 3
Markov chains ergodicity Harris-type conditions 135—162
Markov chains ergodicity proofs 101—107
Markov chains functions, analytical properties 78—79
Markov chains in communication systems 501
Markov chains in positive octants 369—393
Markov chains in random environments 135 165—178
Markov chains limit behavior, classification 424—425
Markov chains mappings 250 264—265
Markov chains non-periodicity 5
Markov chains notation issues 417—418
Markov chains on entire line, transition phenomena 451—463
Markov chains on half-lines 284—289
Markov chains on half-lines ergodicity 284—285
Markov chains on half-lines positive 417—425
Markov chains on half-lines rates of convergence 285—286
Markov chains on half-lines stability theorems 287—289
Markov chains on real lines 289—293
Markov chains positivity conditions 36—49
Markov chains positivity conditions, approaching times method 45—49
Markov chains positivity conditions, Lyapunov method 37—43
Markov chains state space transformations 56
Markov chains stationary distributions 291—292
Markov chains stationary distributions moments 425
Markov chains trajectory deviations, probability estimates 32—35
Markov chains transition phenomena, theorems 420—424 425—450
Markov chains transition probabilities 168 339
Markov chains, almost homogeneous 295
Markov chains, asymptotically homogeneous 248 279 281 370 501
Markov chains, asymptotically homogeneous, large deviations 302—316
Markov chains, asymptotically homogeneous, stability 288
Markov chains, countable 78—108
Markov chains, embedded 408
Markov chains, embedded, ergodicity 236—240
Markov chains, embedded, stopping times 237
Markov chains, extended, construction 7—8
Markov chains, Harris irreducible, theorems 1—53
Markov chains, Harris recurrent 4 548
Markov chains, Harris recurrent, theorems 5
Markov chains, irreducible 12
Markov chains, multi-dimensional 130
Markov chains, multi-dimensional, ergodicity 243—463
Markov chains, multi-dimensional, in communication systems 501
Markov chains, multi-dimensional, non-recurrence 248
Markov chains, multi-dimensional, positive recurrence 245—275
Markov chains, multi-dimensional, stability 243—463
Markov chains, non-Harris irreducible, theorems 55—133
Markov chains, one-dimensional, characteristics 277—316
Markov chains, one-dimensional, transition phenomena 417—463
Markov chains, partially homogeneous 321
Markov chains, partially homogeneous, large deviations 293—301
Markov chains, two-dimensional ergodicity 317—368
Markov chains, two-dimensional, moment conditions for ergodicity 257—272
Markov chains, two-dimensional, stability 317—368
Markov chains, uniformly recurrent 4
Markov diffusion process 223
Markov processes see also diffusion processes; jump processes; semi-Markov processes
Markov processes applications 225
Markov processes asymptotic stochastic continuity 226—230
Markov processes ergodicity 243—463
Markov processes jump 395—416
Markov processes matrices 396—397
Markov processes stability 243—463
Markov processes transition functions 226
Markov processes, continuous-time, ergodicity 225—230 395
Martingales 213 397 469
Martingales differences 477
Matrices in Markov processes 396—397
Mc see Markov chains
Moments bounds 485—496
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