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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 |
Moments estimates 467—500
Moments of large deviations, estimates 467—500
Moments random variables, estimates 467—484
Moments random walks 370—371
Multi-dimensional processes 277—316
Multiple access synchronous broadcasting channel, use of term 539
Non-Harris chains ergodicity conditions 246 284
Non-Harris chains ergodicity studies 55—56
Non-Harris chains examples 57—60
Non-Harris chains phenomena 76
Non-Harris chains positivity conditions 62
Normal law, convergence to 444—450
Notation, Markov chains, issues 417—418
Octants transitory 389
Octants, definition 278
Octants, ergodic 389
Octants, weakly negative 389
Ordering operator, definition 82
Orthants, definition 278
Periodicity, stochastically recursive sequences 148
Planes see also hyperplanes
Planes entire, random walks in 357—366
Planes, half, random walks in 344—357
Planes, infinitely distant 372
Poisson distributions 282 412 536
Poisson processes 549
Poisson processes, independent 505
Poisson processes, input 515—516 523 536
Poisson processes, oscillating 226
Poisson processes, point 411
Polling rings, use of term 506
Polling systems 504—529
Polling systems convergence rates, estimates 521—522
Polling systems cycles, duration 526
Polling systems ergodicity 506—521 549—543
Polling systems of first kind 504—522
Polling systems of second kind 523—514
Polling systems, cyclic, ergodicity 506
Positive recurrence, Markov chains, multi-dimensional 245—275
Positivity conditions, approaching times method 45—49
Positivity conditions, Lyapunov method 37—43
Positivity conditions, Markov chains 36—49
Positivity conditions, non-Harris chains 62
Probabilities see also transition probabilities
Probabilities convergence 90—91
Probabilities of large deviations 293—316 521—522
Probabilities of large deviations, estimates 467—500
Probabilities taboo 28 37
Probabilities, boundedness in 137 209—215
Probabilities, invariant 5
Probabilities, Markov chains, one-dimensional 277—316
Probabilities, random variables, estimates 467—484
Probabilities, random walks, bounds 485—496
Probabilities, random walks, estimates 467—500
Probability densities 69
Probability distributions, invariant 17
Processes see also diffusion processes; jump processes; Markov processes; Poisson processes; semi-Markov processes; stochastic processes; Wiener processes
Processes, Asmussen 236
Processes, asymptotically homogeneous, regular 318—319
Processes, embedded, applications 236
Processes, ergodic 348
Processes, induced 335
Processes, multi-dimensional 277—316
Prokhorov theorem 115
Quadrants, definition 278
Quadrants, ergodic 374
Quadrants, transitory 373—374 387
Quadrants, weakly negative 372—373
Queueing networks see also communication networks; Jackson networks
Queueing networks ergodicity 501—558
Queueing networks historical background 502
Queueing networks stability 501—558
Queueing networks, multi-channel 196
Queueing networks, use of term 501—502
Queueing systems see also polling systems
Queueing systems behaviour, under heavy traffic conditions 424
Queueing systems communication channels 538—544
Queueing systems continuous-time processes 240
Queueing systems service times 503
Queueing systems, inter-arrival times 503
Queueing systems, multi-server 502
Queueing systems, one-channel, virtual waiting time 223
Random access communication convergence rates, estimates 546
Random access communication ergodicity 539—531
Random access communication systems 537—547
Random transformations, and ergodicity conditions 78—108
Random variables Gamma-distribution 452 456
Random variables moments, estimates 467—484
Random variables notation 485—486
Random variables probabilities, estimates 467—484
Random variables recurrence time 493—496
Random variables sums, in random walk generation 485—496
Random variables, class of 16
Random variables, random, uniformly integrable 331
Random walks bounded jumps 248
Random walks classification 278—279
Random walks ergodicity, conditions 369—372
Random walks in entire plane 357—366
Random walks in entire plane, minimum conditions 369
Random walks in half-plane 344—357
Random walks in positive octant, d-dimensional 388—393
Random walks in positive octant, three-dimensional 370—388
Random walks in strips 366—368
Random walks moments bounds 485—496
Random walks moments conditions 370—371
Random walks moments estimates 467—500
Random walks over three-dimensional cones 262
Random walks probabilities bounds 485—496
Random walks probabilities, estimates 467—500
Random walks regularity conditions 370—371
Random walks spiral motion 383 387—388 389 390
Random walks stationary distributions 543
Random walks stopping times 468
Random walks trajectories 102
Random walks, asymptotically homogeneous 279—281 285—286
Random walks, asymptotically homogeneous, ergodicity 317—343
Random walks, asymptotically homogeneous, in positive quadrant 317—344
Random walks, asymptotically homogeneous, stability 343—344
Random walks, homogeneous 293
Random walks, marked Markov, estimates 496—500
Random walks, multi-dimensional 277—274
Random walks, multi-dimensional, ergodicity 281—282 503
Random walks, non-ergodic 96 324
Random walks, one-dimensional, stationary distributions 371
Random walks, oscillating 58 208
Random walks, oscillating, ergodicity 72
Random walks, oscillating, large deviations 292
Random walks, oscillating, stationary distributions 290—291
Random walks, partially homogeneous 280—281 285—286
Random walks, partially spatially homogeneous 266—269 375
Random walks, regular 263 345
RC see recursive chains
Real lines, Markov chains on 289—293
Recurrence time, random variables 493—496
Recursive chains see also stochastically recursive sequences
Recursive chains and stationary sequences 180—181
Recursive chains ergodicity 178—195
Recursive chains ergodicity general criteria 178—195
Recursive chains ergodicity sufficient conditions 184—195
Recursive chains ergodicity under non-stationary control 184
Recursive chains governing sequences, independent elements 231—236
Recursive chains in communication networks 166—167
Recursive chains in communication systems 501
Recursive chains properties 165—175
| Recursive chains reduction, to stochastically recursive sequences 174—178
Recursive chains stability theorem 137
Recursive chains stationary distributions, stability 216—220
Recursive chains strong coupling-convergence 233
Recursive chains, definitions 135—137 165—174
Recursive chains, embedded, definitions 230—231
Recursive chains, embedded, ergodicity 230—241
Recursive chains, embedded, example processes 240—241
Recursive chains, embedded, stationary control 239—240
Recursive chains, homogeneous 178
refinements 485—496
Regularity conditions, random walks 370—371
Renewal Theorem 488
Renovation events existence 503
Renovation events stochastically recursive sequences 224
Renovation events structure 195—198
Renovation events, definition 140
Renovation events, non-stationary 142 145—147 151
Renovation events, stationary 150
Renovation events, stationary, conditions 195—215
Renovation events, stationary, positive 149 155 158 239
Renovation events, stationary, sequences 152—155
Rosenthal — Burkholder moment inequality 469
Semi-Markov processes ergodic theory 237
Semi-Markov processes studies 237
Semimartingales 206
SMPP see stationary marked point processes
Solutions, stabilization of 400
SRS see stochastically recursive sequences
Stability diffusion processes, multidimensional 395—416
Stability general theorems 1—241
Stability of stationary distributions 49—53 413—416
Stability of stationary distributions, recursive chains 216—220
Stability theorems, diffusion processes 413
Stability theorems, Harris chains 52—53
Stability, arbitrary Markov chains 130—133
Stability, communication networks 465—558
Stability, Feller Markov chains, general theorems 114—118
Stability, Markov chains on half-lines 287—289
Stability, Markov chains, multi-dimensional 243—463
Stability, Markov chains, one-dimensional 277—316
Stability, Markov chains, two-dimensional 317—368
Stability, Markov processes 243—463
Stability, queueing networks 501—558
Stability, random walks, asymptotically homogeneous 343—344
Stability, random walks, three-dimensional 388
Stationary distributions approximations 417—463
Stationary distributions diffusion processes 410—411
Stationary distributions limit theorems 418
Stationary distributions stability 49—53 413—416
Stationary distributions stability, recursive chains 216—220
Stationary distributions, Markov chains 291—292
Stationary distributions, Markov chains moments 425
Stationary distributions, random walks 543
Stationary distributions, random walks, one-dimensional 371
Stationary distributions, random walks, oscillating 291—291
Stationary distributions, two-dimensional 373
Stationary marked point processes, definitions 239
Stationary sequences and recursive chains 180—181
Stationary sequences conditions 140
Stationary sequences renovation events 152—155
Stationary sequences, arbitrary 142 163—164
Stochastic continuity in variation 226
Stochastic continuity, asymptotic 226
Stochastic processes ergodicity, in continuous and discrete time 221—241
Stochastic processes in communication systems 501
Stochastically recursive sequences 135—226
Stochastically recursive sequences and generating transformations 162—165
Stochastically recursive sequences and Markov chains, compared 139—140
Stochastically recursive sequences and Markov chains, ergodicity conditions 156—161
Stochastically recursive sequences convergence rates, estimates 162
Stochastically recursive sequences coupling-convergence 148—149 152—154
Stochastically recursive sequences coupling-convergence, strong 149—152 154—155 158
Stochastically recursive sequences ergodicity conditions 136 224
Stochastically recursive sequences ergodicity conditions, Harris-type 135—162
Stochastically recursive sequences in communication systems 501
Stochastically recursive sequences renovation events 140 224
Stochastically recursive sequences with enriched control 137
Stochastically recursive sequences, continuous and discrete time 221—225
Stochastically recursive sequences, definitions 135 139
Stochastically recursive sequences, embedded 240—241
Stochastically recursive sequences, generalized 223
Stochastically recursive sequences, Harris type 165
Stochastically recursive sequences, homogeneous 175
Stochastically recursive sequences, non-stationary, ergodicity 152—154
Stochastically recursive sequences, periodicity 148
Stochastically recursive sequences, real-valued, V-inducing sequences 198—206
Stochastically recursive sequences, recursive chain reduction to 174—178
Stochastically recursive sequences, use of term 139
Stopping times, definition 42
Stopping times, Markov 327 332 334
Stopping times, random walks 468
Strips, random walks in 366—368
Strong coupling-convergence recursive chains 233
Strong coupling-convergence stochastically recursive sequences 149—152 154—155 158
Supermartingales 206
Synchronous broadcast channels, multiple access 166
Taboo probabilities 28 37
Taylor expansion 449
Taylor formula 127 129 309 428 432 435 457
Test functions 245
Test functions mean drift 436 438
Time processes, stochastic, ergodicity 221—241
Token polling rings, use of term 506
Trajectories cycles 34
Trajectories deviations 32—35
Transformations see also Lipschitz transformations
Transformations analytical properties, stochastically recursive sequences 162—165
Transformations contraction 136
Transformations Lipschitz contracting, ergodicity 82—91
Transformations, monotonic, ergodicity conditions 79—82
Transformations, random, and ergodicity conditions 78—108
Transition functions 57
Transition functions as differential equation solution 398
Transition functions Markov processes 226
Transition functions, random 221
Transition kernels continuity 67—78
Transition kernels positivity 67—78
Transition kernels, weak 418
Transition phenomena Markov chains classification 451—463
Transition phenomena Markov chains theorems 420—424 425—450
Transition phenomena Markov chains, one-dimensional 417—463
Transition probabilities 49 399
Transition probabilities and ergodicity 67—78
Transition probabilities continuity 53
Transition probabilities contraction property 57
Transition probabilities convergence, conditions 61
Transition probabilities densities 287
Transition probabilities tightness 62
Transition probabilities weak convergence 57—60
Transition probabilities, continuous 115
Transition probabilities, Markov chains 168 339
V-inducing sequences construction 137
V-inducing sequences, existence of 197—198
V-inducing sequences, existence of, arbitrary state space 206—208
V-inducing sequences, existence of, conditions 198—206 214—215
Vectors convergence 79—80
Vectors diffusion processes 404
Vectors, multi-dimensional 277—278
Vectors, waiting time 196
Virtual waiting time in queueing systems 223 241
von Bahr — Esseen moment inequality 469
Wald identity 234 270 332 468
Wiener processes 223 407 411
Wiener processes, d-dimensional 396
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