Markov Chains
This 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 compa
1 online resource (389 pages).
9780080880228, 0080880223
1048594391
Front Cover; Markov Chains; Copyright Page; CONTENTS; PREFACE TO FIRST EDITION; PREFACE TO SECOND EDITION; INTERDEPENDENCE GUIDE; CHAPTER 0. PRELIMINARIES; 1. Notation; 2. Martingales; 3. The monotone class theorem; 4. Topological spaces and groups; CHAPTER 1. TRANSITION PROBABILITIES. MARKOV CHAINS; 1. Kernels. Transition probabilities; 2. Homogeneous Markov chains; 3. Stopping times. Strong Markov property; 4. Random walks on groups and homogeneous spaces; 5. Analytical properties of integral kernels; CHAPTER 2. POTENTIAL THEORY; 1. Superharmonic functions and the maximum principle. 2. Reduced functions and balayage3. Equilibrium, invariant events and transient sets; 4. Invariant and excessive measures; 5. Randomized stopping times and filling scheme; 6. Resolvents; CHAPTER 3. TRANSIENCE AND RECURRENCE; 1. Discrete Markov chains; 2. Irreducible chains and Harris chains; 3. Topological recurrence of random walks; 4. Recurrence criteria for random walks and applications; CHAPTER 4. POINTWISE ERGODIC THEORY; 1. Preliminaries; 2. Maximal ergodic lemma. Hopf's decomposition; 3. The Chacon-Ornstein theorem for conservative contractions; 4. Applications to Harris chains. 5. Brunel's lemma and the general Chacon-Ornstein theorem6. Subadditive ergodic theory; CHAPTER 5. TRANSIENT RANDOM WALKS. RENEWAL THEORY; 1. The theorem of Choquet and Deny; 2. General lemmas; 3. The renewal theorem for the groups R and Z; 4. The renewal theorem; 5. Refinements and applications; CHAPTER 6. ERGODIC THEORY OF HARRIS CHAINS; 1. The zero-two laws; 2. Cyclic classes and limit theorems for Harris chains; 3. Quasi-compact transition probabilities and strong ergodic theorem; 4. Special functions; 5. Potential kernels; 6. The ratio-limit theorem; CHAPTER 7. MARTIN BOUNDARY. 1. Regular functions2. Convergence to the boundary; 3. Integral representation of harmonic functions; CHAPTER 8. POTENTIAL THEORY FOR HARRIS CHAINS; 1. Harris chains and duality; 2. Equilibrium, balayage and maximum principles; 3. Normal chains; 4. Feller chains and recurrent boundary theory; CHAPTER 9. RECURRENT RANDOM WALKS; 1. Preliminaries; 2. Normality and potential kernels; 3. Martin boundary; 4. Renewal theory; CHAPTER 10. CONSTRUCTION OF MARKOV CHAINS AND RESOLVENTS; 1. Preliminaries and bounded kernels; 2. The reinforced principle. Construction of transient Markov chains. 3. The semi-complete maximum principleNOTES AND COMMENTS; REFERENCES; INDEX OF NOTATION; INDEX OF TERMS