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Markov chains : analytic and Monte Carlo computations /
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 th...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Chichester, West Sussex :
Wiley-Dunod,
2014.
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| Colección: | Wiley series in probability and statistics.
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| Temas: | |
| Acceso en línea: | Texto completo Texto completo |
Tabla de Contenidos:
- Cover; Title Page; Copyright; Contents; Preface; List of Figures; Nomenclature; Introduction; Chapter 1 First steps; 1.1 Preliminaries; 1.2 First properties of Markov chains; 1.2.1 Markov chains, finite-dimensional marginals, and laws; 1.2.2 Transition matrix action and matrix notation; 1.2.3 Random recursion and simulation; 1.2.4 Recursion for the instantaneous laws, invariant laws; 1.3 Natural duality: algebraic approach; 1.3.1 Complex eigenvalues and spectrum; 1.3.2 Doeblin condition and strong irreducibility; 1.3.3 Finite state space Markov chains; 1.4 Detailed examples.
- 2.3 Detailed examples2.3.1 Gambler's ruin; 2.3.2 Unilateral hitting time for a random walk; 2.3.3 Exit time from a box; 2.3.4 Branching process; 2.3.5 Word search; Exercises; Chapter 3 Transience and recurrence; 3.1 Sample paths and state space; 3.1.1 Communication and closed irreducible classes; 3.1.2 Transience and recurrence, recurrent class decomposition; 3.1.3 Detailed examples; 3.2 Invariant measures and recurrence; 3.2.1 Invariant laws and measures; 3.2.2 Canonical invariant measure; 3.2.3 Positive recurrence, invariant law criterion; 3.2.4 Detailed examples; 3.3 Complements.
- 3.3.1 Hitting times and superharmonic functions3.3.2 Lyapunov functions; 3.3.3 Time reversal, reversibility, and adjoint chain; 3.3.4 Birth-and-death chains; Exercises; Chapter 4 Long-time behavior; 4.1 Path regeneration and convergence; 4.1.1 Pointwise ergodic theorem, extensions; 4.1.2 Central limit theorem for Markov chains; 4.1.3 Detailed examples; 4.2 Long-time behavior of the instantaneous laws; 4.2.1 Period and aperiodic classes; 4.2.2 Coupling of Markov chains and convergence in law; 4.2.3 Detailed examples; 4.3 Elements on the rate of convergence for laws.
- 4.3.1 The Hilbert space framework4.3.2 Dirichlet form, spectral gap, and exponential bounds; 4.3.3 Spectral theory for reversible matrices; 4.3.4 Continuous-time Markov chains; Exercises; Chapter 5 Monte Carlo methods; 5.1 Approximate solution of the Dirichlet problem; 5.1.1 General principles; 5.1.2 Heat equation in equilibrium; 5.1.3 Heat equation out of equilibrium; 5.1.4 Parabolic partial differential equations; 5.2 Invariant law simulation; 5.2.1 Monte Carlo methods and ergodic theorems; 5.2.2 Metropolis algorithm, Gibbs law, and simulated annealing.