Preface． ．
Chapter 1 Random Walks A Good Place to Begin
1．1．Nearest Neighbor Random Wlalks on Z．
1．1．1．Distribution at Time n
1．1．2．Passage Times via the Reflection Principle．
1．1．3．Some Related Computations
1．1．4．Time of First Return
1．1．5．Passage Times via Functional Equations
1．2．Recurrence Properties of Random Walks
1．2．1．Random Walks on Zd
1．2．2．An Elementary Recurrence Criterion
1．2．3．Recurrence of Symmetric Random Walk in Zz
1．2．4．nansience in Z3
1．3．Exercises
Chapter 2 Doeblin’S Theory for Markov Chains
2．1．Some Generalities
2．1．1．Existence of Markov Chains
2．1．2．Transion Probabilities＆Probability Vectors
2．1．3．nansition Probabilities and Functions．
2．1．4．The Markov Property
2．2．Doeblin’S Theory．
2．2．1．Doeblin’S Basic Theorem
2．2．2．A Couple of Extensions
2．3．Elements of Ergodic Theory
2．3．1．The Mean Ergodic Theorem
2．3．2．Return Times
2．3．3．Identification of π
2．4．Exercises
Chapter 3 More about the Ergodic Theory of Markov Chains
3．1．Classification of States
3．1．1．Classification，Recurrence，and Transience
3．1．2．Criteria for Recurrence and Transmnge
3．1．3．Periodicity．
3．2．Ergodic Theory without Doeblin
3．2．1．Convergence of Matrices．
3．2．2．Ab el Convergence
3．2．3．Structure of Stationary Distributions
3．2．4．A Small Improvement
3．2．5．The Mcan Ergodic Theorem Again
3．2．6．A Refinement in The Aperiodic Case
3．2．7．Periodic Structure
3．3．Exercises
Chapter 4 Markov Processes in Continuous Time
4．1．Poisson Processes．
4．1．1．The Simple Poisson Process．
4．1．2．Compound Poisson Processes on Z
4．2．Markov Processes with Bounded Rates
4．2．1．Basic Construction
4．2．2．The Markov Property
4．2．3．The Q—Matrix and Kolmogorov’S Backward Equation．
4．2．4．Kolmogorov’S Forward Equation
4．2．5．Solving Kolmogorov’S Equation
4．2．6．A Markov Process from its Infinitesimal Characteristics
4．3．Unbounded Rates
4．3．1．Explosion
4．3．2．Criteria for Non．explosion or Explosion
4．3．3．What to Do When Explosion Occurs．
4．4．Ergodic Properties．
4．4．1．Classification of States．
4．4．2．Stationary Measures and Limit Theorems
4．4．3．Interpreting πii．
4．5．Exercises
Chapter 5 Reversible Markov Proeesses
5．1．R，eversible Markov Chains
5．1．1．Reversibility from Invariance
5．1．2．Measurements in Quadratic Mean
5．1．3．The Spectral Gap．
5．1．4．Reversibility and Periodicity
5．1．5．Relation to Convergence in Variation
5．2．Dirichlet Forms and Estimation of β
5．2．1．The Dirichlet Fo·rm and Poincar4’S Inequality，
5．2．2．Estimating β+
5．2．3．Estimating β-
5．3．Reversible Markov Processes in Continuous Time
5．3．1．Criterion for Reversibility
5．3．2．Convergence in L2(π) for Bounded Rates
5．3．3．L2(π)Convergence Rate in General
……
Chapter 6 Some Mild Measure Theory
Notation
References
Index