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Markov Chains : Analytic and Monte Carlo Computations
发布日期:2015-12-11  浏览

Markov Chains : Analytic and Monte Carlo Computations

[Book Description]

"Markov Chains: Analytic and Monte Carlo Computations" introduces the main notions related to Markov chains and provides explanations on how to characterize, simulate, and recognize them. Starting with basic notions, this book leads progressively to advanced and recent topics in the field, allowing the reader to master the main aspects of the classical theory. This book also features: Numerous exercises with solutions as well as extended case studies. A detailed and rigorous presentation of Markov chains with discrete time and state space. An appendix presenting probabilistic notions that are necessary to the reader, as well as giving more advanced measure-theoretic notions.

[Table of Contents]
 
Preface                                            ix
        List of Figures                            xi
Nomenclature                                       xiii
Introduction                                       xv
    1 First steps                                  1   (46)
      1.1 Preliminaries                            1   (1)
      1.2 First properties of Markov chains        2   (9)
        1.2.1 Markov chains, finite-dimensional    2   (3)
        marginals, and laws
        1.2.2 Transition matrix action and         5   (4)
        matrix notation
        1.2.3 Random recursion and simulation      9   (1)
        1.2.4 Recursion for the instantaneous      10  (1)
        laws, invariant laws
      1.3 Natural duality: algebraic approach      11  (10)
        1.3.1 Complex eigenvalues and spectrum     11  (4)
        1.3.2 Doeblin condition and strong         15  (2)
        irreducibility
        1.3.3 Finite state space Markov chains     17  (4)
      1.4 Detailed examples                        21  (26)
        1.4.1 Random walk on a network             21  (1)
        1.4.2 Gambler's ruin                       22  (3)
        1.4.3 Branching process: evolution of a    25  (2)
        population
        1.4.4 Ehrenfest's Urn                      27  (6)
        1.4.5 Renewal process                      33  (3)
        1.4.6 Word search in a character chain     36  (2)
        1.4.7 Product chain                        38  (2)
        Exercises                                  40  (7)
    2 Past, present, and future                    47  (32)
      2.1 Markov property and its extensions       47  (4)
        2.1.1 Past σ-field, filtration,      47  (1)
        and translation operators
        2.1.2 Markov property                      48  (2)
        2.1.3 Stopping times and strong Markov     50  (1)
        property
      2.2 Hitting times and distribution           51  (9)
        2.2.1 Hitting times, induced chain, and    51  (2)
        hitting distribution
        2.2.2 "One step forward" method,           53  (7)
        Dirichlet problem
      2.3 Detailed examples                        60  (19)
        2.3.1 Gambler's ruin                       60  (4)
        2.3.2 Unilateral hitting time for a        64  (3)
        random walk
        2.3.3 Exit time from a box                 67  (1)
        2.3.4 Branching process                    67  (4)
        2.3.5 Word search                          71  (2)
        Exercises                                  73  (6)
    3 Transience and recurrence                    79  (40)
      3.1 Sample paths and state space             79  (8)
        3.1.1 Communication and closed             79  (1)
        irreducible classes
        3.1.2 Transience and recurrence,           80  (3)
        recurrent class decomposition
        3.1.3 Detailed examples                    83  (4)
      3.2 Invariant measures and recurrence        87  (10)
        3.2.1 Invariant laws and measures          87  (2)
        3.2.2 Canonical invariant measure          89  (2)
        3.2.3 Positive recurrence, invariant       91  (2)
        law criterion
        3.2.4 Detailed examples                    93  (4)
      3.3 Complements                              97  (22)
        3.3.1 Hitting times and superharmonic      97  (2)
        functions
        3.3.2 Lyapunov functions                   99  (6)
        3.3.3 Time reversal, reversibility, and    105 (3)
        adjoint chain
        3.3.4 Birth-and-death chains               108 (3)
        Exercises                                  111 (8)
    4 Long-time behavior                           119 (36)
      4.1 Path regeneration and convergence        119 (9)
        4.1.1 Pointwise ergodic theorem,           120 (4)
        extensions
        4.1.2 Central limit theorem for Markov     124 (2)
        chains
        4.1.3 Detailed examples                    126 (2)
      4.2 Long-time behavior of the                128 (12)
      instantaneous laws
        4.2.1 Period and aperiodic classes         128 (4)
        4.2.2 Coupling of Markov chains and        132 (7)
        convergence in law
        4.2.3 Detailed examples                    139 (1)
      4.3 Elements on the rate of convergence      140 (15)
      for laws
        4.3.1 The Hilbert space framework          140 (3)
        4.3.2 Dirichlet form, spectral gap, and    143 (3)
        exponential bounds
        4.3.3 Spectral theory for reversible       146 (3)
        matrices
        4.3.4 Continuous-time Markov chains        149 (1)
        Exercises                                  150 (5)
    5 Monte Carlo methods                          155 (16)
      5.1 Approximate solution of the Dirichlet    155 (7)
      problem
        5.1.1 General principles                   155 (1)
        5.1.2 Heat equation in equilibrium         156 (2)
        5.1.3 Heat equation out of equilibrium     158 (1)
        5.1.4 Parabolic partial differential       159 (3)
        equations
      5.2 Invariant law simulation                 162 (9)
        5.2.1 Monte Carlo methods and ergodic      162 (1)
        theorems
        5.2.2 Metropolis algorithm, Gibbs law,     163 (3)
        and simulated annealing
        5.2.3 Exact simulation and backward        166 (5)
        recursion
  Appendix A Complements                           171 (24)
      A.1 Basic probabilistic notions              171 (6)
        A.1.1 Discrete random variable,            171 (4)
        expectation, and generating function
        A.1.2 Conditional probabilities and        175 (2)
        independence
      A.2 Discrete measure convergence             177 (6)
        A.2.1 Total variation norm and maximal     177 (3)
        coupling
        A.2.2 Duality between measures and         180 (2)
        functions
        A.2.3 Weak convergence of laws and         182 (1)
        convergence in law
      A.3 Measure-theoretic framework              183 (12)
        A.3.1 Probability spaces                   183 (2)
        A.3.2 Measurable spaces and functions:     185 (1)
        signed and nonnegative
        A.3.3 Random variables, their laws, and    186 (6)
        expectations
        A.3.4 Random sequences and Kolmogorov      192 (3)
        extension theorem
References                                         195 (2)
Solutions for the exercises                        197 (32)
Index                                              229

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