 | Menshikov M. Non-homogeneous random walks: lyapunov function methods for near-critical stochastic systems / M.Menshikov, S.Popov, A.Wade. - Cambridge: Cambridge University Press, 2017. - xviii, 363 p. - (Cambridge tracts in mathematics; 209.). - Bibliogr.: p. 344-360. - Ind.: p.361-363.
- ISBN 978-1-107-02669-8
Шифр: (Pr 1208/209) 02
|
Preface ....................................................... vii
Notation ....................................................... xv
1 Introduction ................................................. 1
1.1 Random Walks ............................................ 1
1.2 Simple Random Walk ...................................... 2
1.3 Lamperti's Problem ...................................... 4
1.4 General Random Walk ..................................... 7
1.5 Recurrence and Transience ............................... 9
1.6 Angular Asymptotics .................................... 13
1.7 Centrally Biased Random Walks .......................... 13
Bibliographical Notes ....................................... 16
2 Semimartingale Approach and Markov Chains ................... 21
2.1 Definitions ............................................ 21
2.2 An Introductory Example ................................ 28
2.3 Fundamental Semimartingale Facts ....................... 31
2.4 Displacement and Exit Estimates ........................ 38
2.5 Recurrence and Transience Criteria for Markov Chains ... 48
2.6 Expectations of Hitting Times and Positive
Recurrence ............................................. 60
2.7 Moments of Hitting Times ............................... 70
2.8 Growth Bounds on Trajectories .......................... 77
Bibliograpical Notes ........................................ 84
3 Lamperti's Problem .......................................... 91
3.1 Introduction ........................................... 91
3.2 Markovian Case ......................................... 93
3.3 General Case ........................................... 96
3.4 Lyapunov Functions .................................... 102
3.5 Recurrence Classification ............................. 107
3.6 Irreducibility and Regeneration ....................... 115
3.7 Moments and Tails of Passage Times .................... 126
3.8 Excursion Durations and Maxima ........................ 131
3.9 Almost-Sure Bounds on Trajectories .................... 136
3.10 Transient Theory in the Critical Case ................. 140
3.11 Nullity and Weak Limits ............................... 147
3.12 Supercritical Case .................................... 153
3.13 Proofs for the Markovian Case ......................... 163
Bibliographical Notes ...................................... 164
4 Many-Dimensional Random Walks .............................. 173
4.1 Introduction .......................................... 173
4.2 Elliptic Random Walks ................................. 190
4.3 Controlled Driftless Random Walks ..................... 194
4.4 Centrally Biased Random Walks ......................... 201
4.5 Range and Local Time of Many-Dimensional
Martingales ........................................... 215
Bibliographical Notes ...................................... 219
5 Heavy Tails ................................................ 224
5.1 Chapter Overview ...................................... 224
5.2 Directional Transience ................................ 225
5.3 Oscillating Random Walk ............................... 239
Bibliographical Notes ...................................... 255
6 Further Applications ....................................... 258
6.1 Random Walk in Random Environment ..................... 258
6.2 Random Strings in Random Environment .................. 273
6.3 Stochastic Billiards .................................. 281
6.4 Exclusion and Voter Models ............................ 287
Bibliographical Notes ...................................... 304
7 Markov Chains in Continuous Time ........................... 311
7.1 Introduction and Notation ............................. 311
7.2 Recurrence and Transience ............................. 314
7.3 Existence and Non-existence of Moments of Passage
Times ................................................. 315
7.4 Explosion and Implosion ............................... 324
7.5 Applications .......................................... 334
Bibliographical Notes ...................................... 337
Glossary of Named Assumptions ................................. 340
References .................................................... 344
Index ......................................................... 361
|
|
