Markov ChainsThis is the revised and augmented edition of a now classic book which is an introduction to sub-Markovian kernels on general measurable spaces and their associated homogeneous Markov chains. The first part, an expository text on the foundations of the subject, is intended for post-graduate students. A study of potential theory, the basic classification of chains according to their asymptotic behaviour and the celebrated Chacon-Ornstein theorem are examined in detail. The second part of the book is at a more advanced level and includes a treatment of random walks on general locally compact abelian groups. Further chapters develop renewal theory, an introduction to Martin boundary and the study of chains recurrent in the Harris sense. Finally, the last chapter deals with the construction of chains starting from a kernel satisfying some kind of maximum principle. |
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Table des matières
| 1 | |
| 8 | |
CHAPTER 2 POTENTIAL THEORY | 40 |
CHAPTER 3 TRANSIENCE AND RECURRENCE | 80 |
CHAPTER 4 POINTWISE ERGODIC THEORY | 117 |
CHAPTER 5 TRANSIENT RANDOM WALKS RENEWAL THEORY | 159 |
CHAPTER 6 ERGODIC THEORY OF HARRIS CHAINS | 186 |
CHAPTER 7 MARTIN BOUNDARY | 233 |
CHAPTER 8 POTENTIAL THEORY FOR HARRIS CHAINS | 261 |
CHAPTER 9 RECURRENT RANDOM WALKS | 286 |
CHAPTER 10 CONSTRUCTION OF MARKOV CHAINS AND RESOLVENTS | 324 |
NOTES AND COMMENTS | 346 |
| 356 | |
| 371 | |
| 372 | |
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Expressions et termes fréquents
abelian groups absorbing set assume asymptotic Borel bounded function bounded harmonic functions bounded measure co-special compact set completes the proof constant converge vaguely Corollary countable decomposition defined Definition denote discrete chains du(x duality easily seen equal ergodic everywhere Exercise exists finite function f g-finite Gf(x group G Haar measure harmonic functions Harris chains hence Hint implies increasing sequence induced inequality integer invariant measure irreducible recurrent isomorphic Lemma Let f Let G limit Markov chain Markov property maximum principle notation operator potential kernel probability measure proof is complete proper kernel Proposition 1.3 Prove q-algebra quasi-compact Radon measure random variables random walk real numbers resolvent equation respect result satisfies sequel special functions strictly positive function subset superharmonic function tends to infinity Theorem 2.1 topology transition probability uniformly vague topology vanishes walk of law zero