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Borovkov A.A. Ч Ergodicity and Stability of Stochastic Processes
Borovkov A.A. Ч Ergodicity and Stability of Stochastic Processes

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Ќазвание: Ergodicity and Stability of Stochastic Processes

јвтор: Borovkov A.A.


Dedicated to the study of ergodicity and stability of stochastic processes this book provides a thorough and up-to-date investigation of these processes. The author is at the forefront of this growing area of research and presents novel results as well as established ideas. The term "stability" is used in this book to describe continuity properties of stationary distributions with respect to small perturbations of local characteristics. Comprising three parts, the first eloquently demonstrates the general theorems of ergodicity and stability for a comprehensive number of classes of Markov chains, stochastically recursive sequences and their generalizations. Expanding on the introduction, the second part considers ergodicity and stability of multi-dimensional Markov chains and Markov processes. For one-dimensional Markov chains special attention is paid to large deviation problems and transient phenomenon. Drawing upon the results presented throughout the book the final part considers their application in establishing conditions of ergodicity in communication and queueing networks. In particular, two types of polling systems are considered; Jackson networks and buffered random access systems related to the ALOHA algorithm. This text will have broad appeal to statisticians and applied researchers seeking new results in the theory of Markov models and their application.

язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1998

 оличество страниц: 585

ƒобавлена в каталог: 02.03.2006

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
ѕредметный указатель
ALOHA (algorithm)      167 502
ALOHA applications      538
Aperiodicity condition, Markov chains      137
Approaching times method      45Ч49 317Ч393
Arzela Ч Ascoli theorem      72
Asmussen processes      236
Autoregression generalized      91Ч97
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 Ч 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      486Ч493
Boundedness, in probabilities      137 209Ч215
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      538Ч544
Communication networks      see also Jackson networks; queueing networks
Communication networks ergodicity      465Ч558
Communication networks stability      465Ч558
Communication networks, recursive chains in      166Ч167
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      536Ч537
Convergence rates polling systems      521Ч522
Convergence rates random access communication systems      547
Convergence rates stochastically recursive sequences      162
Convergence, majorized      52
Coupling method, in theorems      25Ч27
Coupling-convergence      see also strong coupling-convergence
Coupling-convergence stochastically recursive sequences      148Ч149 153Ч154
Cramer transform      91
CramerТs condition      301 325 326 350 474 484 485 488
CramerТs 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      293Ч316 467Ч500 521Ч522
Deviations, large, random walks      285Ч286
Differential equations, stochastic      223 396 411
Differential equations, transition function as solution      398
Diffusion coefficients      398
Diffusion processes density      399
Diffusion processes ergodicity      395Ч411
Diffusion processes ergodicity conditions      400Ч411
Diffusion processes Feller property      414
Diffusion processes stability theorems      414
Diffusion processes stationary distributions      410Ч411
Diffusion processes with reflection      397Ч398
Diffusion processes, asymptotically spatially homogeneous      407Ч410
Diffusion processes, multi-dimensional      416
Diffusion processes, multi-dimensional, ergodicity and stability      395Ч416
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 Ч Wasserstein distance      56
Distributions Levy Ч Prokhorov distance      85 101 131 230
Distributions probability, invariant      17
Distributions rate of convergence      90Ч91
Distributions stationary, stability      49Ч53
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 165Ч178
Equilibrium identities, Markov chains      106Ч114
Ergodicity and transition probabilities      67Ч80
Ergodicity communication networks      465Ч558
Ergodicity conditions      103
Ergodicity conditions and random transformations      78Ч108
Ergodicity conditions for Lipschitz transformations      97Ч102 103
Ergodicity conditions for monotonic transformations      79Ч82
Ergodicity conditions Harris-type      524
Ergodicity conditions necessity      518Ч521
Ergodicity conditions sufficient      184Ч195 501
Ergodicity criteria, in arbitrary dimensions      272Ч274
Ergodicity diffusion processes      395Ч410
Ergodicity diffusion processes conditions      400Ч411
Ergodicity diffusion processes, multi-dimensional      395Ч416
Ergodicity for Lipschitz contracting transformations      82Ч91
Ergodicity general theorems      1Ч241
Ergodicity in the mean      55Ч67
Ergodicity in the mean Ioshida theorem      67
Ergodicity in the mean, definition      60
Ergodicity Jackson networks, open      530Ч536
Ergodicity jump processes      411Ч413
Ergodicity main theorem      11Ч32
Ergodicity Markov chains conditions      11Ч14 118 136 137Ч138
Ergodicity Markov chains early studies      3
Ergodicity Markov chains on half-lines      284Ч286
Ergodicity Markov chains proofs      101Ч107
Ergodicity Markov chains, Harris-type conditions      135Ч162
Ergodicity Markov chains, multi-dimensional      243Ч463
Ergodicity Markov chains, one-dimensional      277Ч316
Ergodicity Markov chains, two-dimensional      257Ч272 317Ч368
Ergodicity Markov processes      243Ч463
Ergodicity Markov processes continuous-time      225Ч230 395
Ergodicity of stochastic processes, in continuous and discrete time      221Ч241
Ergodicity oscillating random walks      72
Ergodicity partial      504
Ergodicity polling systems, of first kind      506Ч521 549Ч543
Ergodicity proofs, problems      246
Ergodicity queueing networks      501Ч558
Ergodicity random walks conditions      369Ч372
Ergodicity random walks, asymptotically homogeneous      317Ч343
Ergodicity random walks, multi-dimensional      281Ч282 503
Ergodicity recursive chains      178Ч195
Ergodicity stochastically recursive sequences, Harris-type conditions      135Ч162
Ergodicity theorems      56 374Ч375
Ergodicity theorems approaches      164Ч165
Ergodicity theorems convergence rates      14 19
Ergodicity theorems, simplified      29Ч30
Ergodicity weak convergence      76
Exponentiality conditions, relaxation      547Ч543
Exponentiality partial      502
Faces, definition      278
Factorization identities      294
Fatou lemma      475
Feller Markov chains stability      288 334
Feller Markov chains stability general theorems      114Ч118 293
Feller property      114
Feller property diffusion processes      414
FIFO services      504
Fokker Ч Planck equation      399
Foster Ч Mousstafa Ч 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      420Ч421 425Ч434
Generalized autoregression      91Ч97
Half-lines, definition      278
Half-lines, infinitely distant      372
Half-lines, Markov chains on      284Ч289
Half-lines, positive, Markov chains on      417Ч425
Harris chains      550 (see also non-Harris chains)
Harris chains ergodicity      390Ч391
Harris chains ergodicity studies      55
Harris chains stability theorems      52Ч53 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
HopfТs theory      168
Hyperplanes, normal vectors      272
Hypersurfaces      252 274
Hypersurfaces tangent hyperplanes      274
Integrals convergence      69
Integrals, Bochner      108
Invariant measures, tightness conditions      118Ч130
Ioshida theorem      67
Jackson networks      529Ч537
Jackson networks convergence rates, estimates      536Ч537
Jackson networks ergodicity      411Ч413
Jackson networks regularity properties      412
Jackson networks stability      522
Jackson networks, closed      529
Jackson networks, open      529Ч531 548
Jackson networks, open, ergodicity      530Ч536
Jump processes, definitions      411
Kantorovich Ч Wasserstein distance between distributions      56
Kolmogorov equation      400
Kolmogorov theorem      187 194 224
Kolmogorov Ч Chapman equation      67 226 399
Kolmogorov, Andrei Nikolaevich (1903Ч1987), 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 Ч 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      97Ч102 103
Lipschitz violation      125
Lyapunov conditions positivity      37Ч43
Lyapunov conditions, use of term      42
Lyapunov exponents, definition      95
Lyapunov functions      282
Lyapunov functions estimates      96
Lyapunov functions existence      252Ч256 267
Lyapunov functions in diffusion processes      400
Lyapunov functions methods      245Ч275
LyapunovТs inequality      478
Majorants stationary      195Ч215 503
Majorants stationary construction      92Ч94
Markov chains      see also Feller Markov chains; Harris chains; non-Harris chains; random walks; recursive chains
Markov chains and stochastically recursive ergodicity conditions      156Ч161
Markov chains and stochastically recursive sequences, compared      139Ч140
Markov chains aperiodicity condition      137
Markov chains arbitrary dimensions, criteria      272Ч274
Markov chains arbitrary, stability      130Ч133
Markov chains characterization      178
Markov chains classification      278Ч279 284
Markov chains convergence rates, estimates      161
Markov chains curvilinear coordinates      249
Markov chains distributions      8Ч11
Markov chains distributions, finite-dimensional      156
Markov chains equilibrium identities      106Ч114
Markov chains ergodicity assumptions      6
Markov chains ergodicity conditions      11Ч14 118 136 137Ч138
Markov chains ergodicity early studies      3
Markov chains ergodicity Harris-type conditions      135Ч162
Markov chains ergodicity proofs      101Ч107
Markov chains functions, analytical properties      78Ч79
Markov chains in communication systems      501
Markov chains in positive octants      369Ч393
Markov chains in random environments      135 165Ч178
Markov chains limit behavior, classification      424Ч425
Markov chains mappings      250 264Ч265
Markov chains non-periodicity      5
Markov chains notation issues      417Ч418
Markov chains on entire line, transition phenomena      451Ч463
Markov chains on half-lines      284Ч289
Markov chains on half-lines ergodicity      284Ч285
Markov chains on half-lines positive      417Ч425
Markov chains on half-lines rates of convergence      285Ч286
Markov chains on half-lines stability theorems      287Ч289
Markov chains on real lines      289Ч293
Markov chains positivity conditions      36Ч49
Markov chains positivity conditions, approaching times method      45Ч49
Markov chains positivity conditions, Lyapunov method      37Ч43
Markov chains state space transformations      56
Markov chains stationary distributions      291Ч292
Markov chains stationary distributions moments      425
Markov chains trajectory deviations, probability estimates      32Ч35
Markov chains transition phenomena, theorems      420Ч424 425Ч450
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      302Ч316
Markov chains, asymptotically homogeneous, stability      288
Markov chains, countable      78Ч108
Markov chains, embedded      408
Markov chains, embedded, ergodicity      236Ч240
Markov chains, embedded, stopping times      237
Markov chains, extended, construction      7Ч8
Markov chains, Harris irreducible, theorems      1Ч53
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      243Ч463
Markov chains, multi-dimensional, in communication systems      501
Markov chains, multi-dimensional, non-recurrence      248
Markov chains, multi-dimensional, positive recurrence      245Ч275
Markov chains, multi-dimensional, stability      243Ч463
Markov chains, non-Harris irreducible, theorems      55Ч133
Markov chains, one-dimensional, characteristics      277Ч316
Markov chains, one-dimensional, transition phenomena      417Ч463
Markov chains, partially homogeneous      321
Markov chains, partially homogeneous, large deviations      293Ч301
Markov chains, two-dimensional ergodicity      317Ч368
Markov chains, two-dimensional, moment conditions for ergodicity      257Ч272
Markov chains, two-dimensional, stability      317Ч368
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      226Ч230
Markov processes ergodicity      243Ч463
Markov processes jump      395Ч416
Markov processes matrices      396Ч397
Markov processes stability      243Ч463
Markov processes transition functions      226
Markov processes, continuous-time, ergodicity      225Ч230 395
Martingales      213 397 469
Martingales differences      477
Matrices in Markov processes      396Ч397
Mc      see Markov chains
Moments bounds      485Ч496
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