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Markov Chains [2018]

1
Introduction
1
Probability Background
3
Markov Chains: Basic Definitions
11
Preliminaries
27
Examples of Markov Chains
39
Gambling Problems
45
Tools for Non-commuting Operators
53
Stopping Times and the Strong Markov Property
61
Multivariate Trace Inequalities
69
Random Walks
75
Approximate Quantum Markov Chains
75
Martingales, Harmonic Functions and Poisson–Dirichlet Problems
89
Discrete-Time Markov Chains
97
Ergodic Theory for Markov Chains
115
First Step Analysis
119
Atomic Chains
145
Markov Chains on a Discrete State Space
147
Classification of States
163
Long-Run Behavior of Markov Chains
165
Convergence of Atomic Markov Chains
189
Branching Processes
191
Small Sets, Irreducibility, and Aperiodicity
211
Continuous-Time Markov Chains
221
Transience, Recurrence, and Harris Recurrence
241
Splitting Construction and Invariant Measures
263
Discrete-Time Martingales
265
Feller and <Emphasis Type="BoldItalic">T</Emphasis>-Kernels
281
Spatial Poisson Processes
289
Rates of Convergence for Atomic Markov Chains
289
Reliability Theory
313
Geometric Recurrence and Regularity
339
Geometric Rates of Convergence
361
(<Emphasis Type="Italic">f</Emphasis>, <Emphasis Type="Italic">r</Emphasis>)-Recurrence and Regularity
385
Subgeometric Rates of Convergence
401
Uniform and <Emphasis Type="Italic">V</Emphasis>-Geometric Ergodicity by Operator Methods
421
Coupling for Irreducible Kernels
455
Convergence in the Wasserstein Distance
489
Central Limit Theorems
523
Spectral Theory
575
Concentration Inequalities
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