Ergodicity and stability of stochastic processes:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English Russian |
| Published: |
Chichester [u.a.]
Wiley
1998
|
| Series: | Wiley series in probability and statistics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIII, 585 S. |
| ISBN: | 0471979139 |
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MARC
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| 100 | 1 | |a Borovkov, Aleksandr A. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Ergodicity and stability of stochastic processes |c A. A. Borovkov |
| 264 | 1 | |a Chichester [u.a.] |b Wiley |c 1998 | |
| 300 | |a XXIII, 585 S. | ||
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| 650 | 4 | |a Ergodic theory | |
| 650 | 4 | |a Stability | |
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| _version_ | 1804126586452049920 |
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| adam_text | IMAGE 1
ERGODICITY AND STABILITY OF
STOCHASTIC PROCESSES
A.A. BOROVKOV
ACADEMY OF SCIENCES SIBERIAN SECTION, RUSSIA
TRANSLATED BY V. YURINSKY
UNIVERSIDADE DA BEIRA INFERIOR, PORTUGAL
JOHN WILEY & SONS
CHICHESTER * NEW YORK * WEINHEIM * BRISBANE * SINGAPORE * TORONTO
IMAGE 2
CONTENTS
AUTHOR S SUMMARY XIII
PREFACE XV
PARTI GENERAL THEOREMS ON ERGODICITY AND STABILITY 1
C H A P T ER 1 GENERAL ERGODICITY AND STABILITY THEOREMS FOR HARRIS
IRREDUCIBLE MARKOV CHAINS 3
§1 INTRODUCTION 3
§2 THE MAIN ERGODICITY THEOREM AND ITS REFINEMENTS 11
§3 ESTIMATES FOR PROBABILITIES OF LARGE DEVIATIONS OF TRAJECTORIES OF A
CHAIN 32
§4 CONDITIONS OF POSITIVITY OF A CHAIN. METHOD OF LYAPUNOV TEST
FUNCTIONS. METHOD OF APPROACHING TIMES 36
4.1. PRELIMINARY REMARKS 36
4.2. THE FIRST CRITERION OF POSITIVITY FOR MARKOV CHAINS. THE LYAPUNOV
METHOD 37
4.3. THE SECOND POSITIVITY CRITERION 43
4.4. ESTIMATES FOR THE DISTRIBUTION OF T{X) IN TERMS OF APPROACHING
TIMES R]{X) (SUFFICIENT CONDITIONS FOR VALIDITYOF (I 3 ,M)) 45
§5 STABILITY OF STATIONARY DISTRIBUTIONS . . .. 49
C H A P T ER 2 ERGODICITY AND STABILITY CONDITIONS FOR MARKOV CHAINS N
OT RELATED TO HARRIS IRREDUCIBILITY 55
§6 INTRODUCTION. ERGODICITY IN THE MEAN . . .. 55
IMAGE 3
VI
CONTENTS
6.1. INTRODUCTION 55
6.2. ERGODICITY IN THE MEAN 61
§7 ERGODICITY IN TERMS OF TRANSITION PROBABILITIES. AN APPROACH BASED ON
POSITIVITY AND CONTINUITY OF TRANSITION KERNELS 67
§8 ERGODICITY CONDITIONS RELATED TO ANALYTICAL PROPERTIES OF RANDOM
TRANSFORMATIONS DEFINING THE CHAIN. REMARKS ON COUNTABLE MARKOV CHAINS
78
8.1. INTRODUCTION 78
8.2. ERGODICITY CONDITIONS FOR MONOTONIC TRANSFORMATIONS. . 79 8.3.
ERGODICITY FOR LIPSCHITZ CONTRACTING TRANSFORMATIONS.. . 82 8.4.
GENERALIZED AUTOREGRESSION 91
8.5. CONDITIONS OF ERGODICITY FOR LIPSCHITZ TRANSFORMATIONS WITH
POSITIVE PROBABILITY OF POSSESSING THE CONTRAC-TION PROPERTY ON
POSITIVE SUBSETS OF STATE SPACE. . 97 8.6. ON POSSIBLE APPROACHES TO
PROVING THE ERGODICITY OF AN
MC 102
8.7. SOME OBSERVATIONS ON COUNTABLE MARKOV CHAINS 104
§9 EQUILIBRIUMIDENTITIES 106
§10 STABILITY CONDITIONS 114
10.1. GENERAL THEOREMS ON STABILITY FOR FELLER MARKOV CHAINS. 114 10.2.
ON TIGHTNESS CONDITIONS FOR INVARIANT MEASURES 118
10.3. STABILITY OF ARBITRARY CHAINS 130
C H A P T ER 3 STOCHASTICALLY RECURSIVE SEQUENCES AND THEIR
GENERALIZATIONS (MARKOV CHAINS IN RANDOM E N V I R O N M E N T S) 135
§11 INTRODUCTION. HARRIS-TYPE ERGODICITY CON DITIONS FOR MARKOV CHAINS
AND STOCHASTI CALLY RECURSIVE SEQUENCES 135
11.1. INTRODUCTION 135
11.2. MARKOV CHAINS 137
11.3. STOCHASTICALLY RECURSIVE SEQUENCES (SRSS) 139
11.4. ERGODICITY OF SRSS FOR NON-STATIONARY GOVERNING SEQUENCES 152
11.5. A COMPARISON OF ERGODICITY CONDITIONS FOR MCS AND SRSS. 156 11.6.
CONVERGENCE RATE ESTIMATES 161
§12 ERGODICITY CONDITIONS FOR STOCHASTICALLY RECURSIVE SEQUENCES IN
TERMS OF ANALYTICAL PROPERTIES OF GENERATING TRANSFORMATIONS 162
IMAGE 4
VU
§13 RECURSIVE CHAINS (MARKOV CHAINS IN RANDOM ENVIRONMENT) 165
13.1. PRINCIPAL DEFINITIONS AND PROPERTIES 165
13.2. REDUCTION OF RCS TO SRSS 174
§14 ERGODICITY OF RECURSIVE CHAINS 178
14.1. GENERAL CRITERIA OF ERGODICITY OF RCS 178
14.2. ERGODICITY OF RCS UNDER NON-STATIONARY CONTROL 184
14.3. SOME SUFFICIENT CONDITIONS OF ERGODICITY 185
§15 CONDITIONS FOR EXISTENCE OF STATIONARY RENOVATION EVENTS. STATIONARY
MAJORANTS. BOUNDEDNESS IN PROBABILITY 195
15.1. THE STRUCTURE OFRENOVATION EVENTS 195
15.2. CONDITIONS OF EXISTENCE OF STATIONARY V-INDUCING EVENTS FOR
REAL-VALUED SRSS 198
15.3. EXISTENCE OF V-INDUCING EVENTS: ARBITRARY STATE SPACE. 206 15.4.
BOUNDEDNESS IN PROBABILITY 209
15.5. ALTERNATIVE CONDITIONS FOR EXISTENCE OF V-INDUCING EVENTS AND
BOUNDEDNESS IN PROBABILITY 214
§16 STABILITY OF STATIONARY DISTRIBUTIONS OF RECURSIVE CHAINS 216
C H A P T ER 4 ERGODICITY OF STOCHASTIC PROCESSES IN CONTINUOUS AND
DISCRETE TIME 221
§17 INTRODUCTION 221
§18 ERGODICITY OF CONTINUOUS-TIME MARKOV PRO CESSES 225
§19 ERGODICITY OF PROCESSES ADMITTING EMBEDDED RECURSIVE CHAINS OR
MARKOV CHAINS . .. 230 19.1. PRINCIPAL DEFINITIONS 230
19.2. ERGODICITY OF PROCESSES WHOSE EMBEDDED RC IS CONTROLLED BY A
SEQUENCE OF INDEPENDENT ELEMENTS. . 231 19.3. ERGODICITY OF PROCESSES
THAT ADMIT EMBEDDED MARKOV CHAINS 236
19.4. ERGODICITY OF PROCESSES WITH AN EMBEDDED RC HAVING STATIONARY
CONTROL 239
19.5. EXAMPLES OF PROCESSES ADMITTING AN EMBEDDED RC. . . 240
IMAGE 5
VLLL
CONTENTS
PART II ERGODICITY AND STABILITY OF MULTI-DIMENSIONAL MARKOV CHAINS AND
MARKOV PROCESSES 243
C H A P T ER 5 CONDITIONS OF POSITIVE RECURRENCE AND ERGODICITY
MULTIDIMENSIONAL MARKOV CHAINS AND THE M E T H OD OF LYAPUNOV FUNCTIONS
245
§20 INTRODUCTION 245
§21 EXISTENCE OF LYAPUNOV FUNCTION IN THEO REM 20.1: CONDITIONS IN
TERMS OF SURFACES . . 252 §22 MOMENT CONDITIONS FOR ERGODICITY OF A
CHAIN IN TWO DIMENSIONS 257
§23 ERGODICITY CRITERIA IN ARBITRARY DIMENSIONS 272
C H A P T ER 6 A DESCRIPTION OF MULTI-DIMENSIONAL PROCESSES TO BE
STUDIED. ERGODICITY, STABILITY, AND PROBABILITIES OF LARGE DEVIATIONS OF
ONE-DIMENSIONAL MARKOV CHAINS 277
§24 MULTI-DIMENSIONAL RANDOM WALKS CONSIDERED IN CHAPTERS 6-8: A
DESCRIPTION 277
§25 MARKOV CHAINS ON A HALF-LINE 284
25.1. ERGODICITY 284
25.2. RATE OF CONVERGENCE AND LARGE DEVIATIONS 285
25.3. THEOREMS OF STABILITY 287
§26 MARKOV CHAINS ON THE ENTIRE REAL LINE . . .. 289 §27 PROBABILITIES
OF LARGE DEVIATIONS 293
27.1. PROBABILITIES OF LARGE DEVIATIONS FOR PARTIALLY HOMOGENEOUS CHAINS
293
27.2. PROBABILITIES OF LARGE DEVIATIONS FOR ASYMPTOTICALLY HOMOGENEOUS
CHAINS 302
C H A P T ER 7 ERGODICITY AND STABILITY OF TWO-DIMENSIONAL MARKOV CHAINS
AND THE M E T H OD OF APPROACHING TIMES 317
§28 ASYMPTOTICALLY HOMOGENEOUS RANDOM WALKS IN THE POSITIVE QUADRANT 317
28.1. ERGODICITY. 317
28.2. STABILITY THEOREMS 343
§29 RANDOM WALKS IN HALF-PLANE 344
§30 RANDOM WALK IN ENTIRE PLANE 357
§31 RANDOM WALK IN A STRIP 366
IMAGE 6
CONTENTS IX
C H A P T ER 8 MARKOV CHAINS IN POSITIVE OCTANTS OF THREE AND MORE
DIMENSIONS AND THE M E T H OD OF APPROACHING TIMES 369
§32 RANDOM WALKS IN THE THREE-DIMENSIONAL POSITIVE OCTANT 370
§33 RANDOM WALKS IN THE POSITIVE D-DIMENSIONAL OCTANT 388
C H A P T ER 9 ERGODICITY AND STABILITY OF MULTI-DIMENSIONAL DIFFUSIONS
AND JUMP MARKOV PROCESSES 395
§34 ERGODICITY OF DIFFUSION PROCESSES 395
34.1. THE FIRST GROUP OF ERGODICITY CONDITIONS 400
34.2. THE SECOND GROUP OF ERGODICITY CONDITIONS 406
§35 ERGODICITY OF JUMP PROCESSES 411
§36 STABILITY OF STATIONARY DISTRIBUTIONS . . .. 413
C H A P T ER 10 TRANSITION PHENOMENA FOR ONE-DIMENSIONAL MARKOV CHAINS:
APPROXIMATION OF STATIONARY DISTRIBUTIONS 417
§37 INTRODUCTION. STATEMENT OF RESULTS FOR CHAINS ON THE POSITIVE
HALF-LINE 417
§38 CONVERGENCE TO GAMMA-DISTRIBUTION. PROOF OF THEOREM 37.2 425
§39 CRITICAL CASE: PROOFS OF THEOREMS 37.3-37.4 . 435 §40 CONVERGENCE TO
NORMAL LAW: PROOF OF THEOREM 37.5 444
§41 CLASSIFICATION OF TRANSITION PHENOMENA FOR MARKOV CHAINS ON THE
ENTIRE LINE 451
P A RT I II AUXILIARY PROPOSITIONS. ERGODICITY AND STABILITY OF QUEUEING
AND COMMUNICATION NETWORKS 465
C H A P T ER 11 ESTIMATES OF MOMENTS AND PROBABILITIES OF LARGE
DEVIATIONS FOR CERTAIN RANDOM WALKS 467
§42 ESTIMATES OF MOMENTS AND PROBABILITIES OF LARGE DEVIATIONS FOR SUMS
OF A RANDOM NUMBER OF RANDOM VARIABLES 467
42.1. INTRODUCTION. A SURVEY OF RESULTS 467
IMAGE 7
X
CONTENTS
42.2. PROOFOF THEOREM 42.1 475
42.3. PROOF OF THEOREM 42.2 478
42.4. PROOFOF LEMMA 42.1 AND THEOREM 42.3 481
42.5. PROOF OF THEOREM 42.4 484
§43 BOUNDS FOR MOMENTS AND PROBABILITIES OF LARGE DEVIATIONS FOR
BOUNDARY FUNCTION ALS OF RANDOM WALKS GENERATED BY SUMS OF RANDOM
VARIABLES. A REFINEMENT OF THE RE NEWAL THEOREM 485
43.1. NOTATION 485
43.2. ASYMPTOTICS OF DISTRIBUTIONS FOR BOUNDARY FUNCTIONALS. 486 43.3.
RECURRENCE TIME 493
43.4. A REFINEMENT OF THE RENEWAL THEOREM 494
§44 ESTIMATES OF MOMENTS AND LARGE DEVIATION PROBABILITIES FOR CERTAIN
BOUNDARY FUNC TIONALS OF MARKED MARKOV RANDOM WALKS . 496
C H A P T ER 12 ERGODICITY AND STABILITY OF QUEUEING AND C O M M U N I C
A T I ON NETWORKS 501
§45 INTRODUCTION 501
§46 POLLING SYSTEMS 504
46.1. POLLING SYSTEMS OF THE FIRST KIND 504
46.2. ERGODICITY OF POLLING SYSTEMS OF THE FIRST KIND 506
46.3. NECESSITY OF ERGODICITY CONDITIONS 518
46.4. ESTIMATES FOR CONVERGENCE RATE AND PROBABILITIES OF LARGE
DEVIATIONS. STABILITY THEOREM 521
46.5. POLLING SYSTEMS OF THE SECOND KIND 523
§47 JACKSON NETWORKS 529
47.1.INTRODUCTION 529
47.2. ERGODICITY OF OPEN JACKSON NETWORKS 530
47.3. ESTIMATES OF CONVERGENCE RATE AND PROBABILITIES OF LARGE
DEVIATIONS. STABILITY 536
§48 RANDOM ACCESS COMMUNICATION SYSTEMS . .. 537 48.1. INTRODUCTION 537
48.2. ERGODICITY. 540
48.3. A SUMCIENT CONDITION OF ERGODICITY UNDER ARBITRARY STATIONARY
CONTROL 544
48.4. ESTIMATES FOR CONVERGENCE RATE AND PROBABILITIES OF LARGE
DEVIATIONS. STABILITY. 546
§49 RELAXATION OF EXPONENTIALITY CONDITIONS . 547
IMAGE 8
CONTENTS XI
49.1. SOME POSSIBLE APPROACHES TO RELAXING EXPONENTIALITY CONDITIONS 547
49.2. ERGODICITY OF POLLING SYSTEMS OF THE FIRST TYPE 549
REFERENCES 559
SUBJECT INDEX 577
|
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| isbn | 0471979139 |
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| spelling | Borovkov, Aleksandr A. Verfasser aut Ergodicity and stability of stochastic processes A. A. Borovkov Chichester [u.a.] Wiley 1998 XXIII, 585 S. txt rdacontent n rdamedia nc rdacarrier Wiley series in probability and statistics Processus stochastiques ram Stabilité ram Théorie ergodique ram Ergodic theory Stability Stochastic processes GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008114883&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Borovkov, Aleksandr A. Ergodicity and stability of stochastic processes Processus stochastiques ram Stabilité ram Théorie ergodique ram Ergodic theory Stability Stochastic processes |
| title | Ergodicity and stability of stochastic processes |
| title_auth | Ergodicity and stability of stochastic processes |
| title_exact_search | Ergodicity and stability of stochastic processes |
| title_full | Ergodicity and stability of stochastic processes A. A. Borovkov |
| title_fullStr | Ergodicity and stability of stochastic processes A. A. Borovkov |
| title_full_unstemmed | Ergodicity and stability of stochastic processes A. A. Borovkov |
| title_short | Ergodicity and stability of stochastic processes |
| title_sort | ergodicity and stability of stochastic processes |
| topic | Processus stochastiques ram Stabilité ram Théorie ergodique ram Ergodic theory Stability Stochastic processes |
| topic_facet | Processus stochastiques Stabilité Théorie ergodique Ergodic theory Stability Stochastic processes |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008114883&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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