國立虎尾科技大學 |

Markov chains

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Markov chains/ by Randal Douc ... [et al.].
other author:	Douc, Randal.
Published:	Cham :Springer International Publishing : : 2018.,
Description:	xviii, 757 p. :ill. (some col.), digital ; : 24 cm.;
Contained By:	Springer eBooks
Subject:	Markov processes. -
Online resource:	https://doi.org/10.1007/978-3-319-97704-1
ISBN:	9783319977041

Markov chains
Markov chains[electronic resource] /by Randal Douc ... [et al.]. - Cham :Springer International Publishing :2018. - xviii, 757 p. :ill. (some col.), digital ;24 cm. - Springer series in operations research and financial engineering,1431-8598. - Springer series in operations research and financial engineering..

Part I Foundations -- Markov Chains: Basic Definitions -- Examples of Markov Chains -- Stopping Times and the Strong Markov Property -- Martingales, Harmonic Functions and Polsson-Dirichlet Problems -- Ergodic Theory for Markov Chains -- Part II Irreducible Chains: Basics -- Atomic Chains -- Markov Chains on a Discrete State Space -- Convergence of Atomic Markov Chains -- Small Sets, Irreducibility and Aperiodicity -- Transience, Recurrence and Harris Recurrence -- Splitting Construction and Invariant Measures -- Feller and T-kernels -- Part III Irreducible Chains: Advanced Topics -- Rates of Convergence for Atomic Markov Chains -- Geometric Recurrence and Regularity -- Geometric Rates of Convergence -- (f, r)-recurrence and Regularity -- Subgeometric Rates of Convergence -- Uniform and V-geometric Ergodicity by Operator Methods -- Coupling for Irreducible Kernels -- Part IV Selected Topics -- Convergence in the Wasserstein Distance -- Central Limit Theorems -- Spectral Theory -- Concentration Inequalities -- Appendices -- A Notations -- B Topology, Measure, and Probability -- C Weak Convergence -- D Total and V-total Variation Distances -- E Martingales -- F Mixing Coefficients -- G Solutions to Selected Exercises.

This book covers the classical theory of Markov chains on general state-spaces as well as many recent developments. The theoretical results are illustrated by simple examples, many of which are taken from Markov Chain Monte Carlo methods. The book is self-contained while all the results are carefully and concisely proven. Bibliographical notes are added at the end of each chapter to provide an overview of the literature. Part I lays the foundations of the theory of Markov chain on general state-spaces. Part II covers the basic theory of irreducible Markov chains starting from the definition of small and petite sets, the characterization of recurrence and transience and culminating in the Harris theorem. Most of the results rely on the splitting technique which allows to reduce the theory of irreducible to a Markov chain with an atom. These two parts can serve as a text on Markov chain theory on general state-spaces. Although the choice of topics is quite different from what is usually covered in a classical Markov chain course, where most of the emphasis is put on countable state space, a graduate student should be able to read almost all of these developments without any mathematical background deeper than that needed to study countable state space (very little measure theory is required) Part III deals with advanced topics on the theory of irreducible Markov chains, covering geometric and subgeometric convergence rates. Special attention is given to obtaining computable convergence bounds using Foster-Lyapunov drift conditions and minorization techniques. Part IV presents selected topics on Markov chains, covering mostly hot recent developments. It represents a biased selection of topics, reflecting the authors own research inclinations. This includes quantitative bounds of convergence in Wasserstein distances, spectral theory of Markov operators, central limit theorems for additive functionals and concentration inequalities. Some of the results in Parts III and IV appear for the first time in book form and some are original.

ISBN: 9783319977041

Standard No.: 10.1007/978-3-319-97704-1doiSubjects--Topical Terms:

527825
Markov processes.

LC Class. No.: QA274.7

Dewey Class. No.: 519.233

Markov chains
LDR:04363nam a2200349 a 4500 001 930581
003 DE-He213
005 20190513105617.0
006 m d
007 cr nn 008maaau
008 190627s2018 gw s 0 eng d
020 $a 9783319977041 $q (electronic bk.)
020 $a 9783319977034 $q (paper)
024 7 $a 10.1007/978-3-319-97704-1 $2 doi
035 $a 978-3-319-97704-1
040 $a GP $c GP
041 0 $a eng
050 4 $a QA274.7
072 7 $a PBT $2 bicssc
072 7 $a MAT029000 $2 bisacsh
072 7 $a PBT $2 thema
072 7 $a PBWL $2 thema
082 0 4 $a 519.233 $2 23
090 $a QA274.7 $b .M346 2018
245 0 0 $a Markov chains $h [electronic resource] / $c by Randal Douc ... [et al.].
260 $a Cham : $c 2018. $b Springer International Publishing : $b Imprint: Springer,
300 $a xviii, 757 p. : $b ill. (some col.), digital ; $c 24 cm.
490 1 $a Springer series in operations research and financial engineering, $x 1431-8598
505 0 $a Part I Foundations -- Markov Chains: Basic Definitions -- Examples of Markov Chains -- Stopping Times and the Strong Markov Property -- Martingales, Harmonic Functions and Polsson-Dirichlet Problems -- Ergodic Theory for Markov Chains -- Part II Irreducible Chains: Basics -- Atomic Chains -- Markov Chains on a Discrete State Space -- Convergence of Atomic Markov Chains -- Small Sets, Irreducibility and Aperiodicity -- Transience, Recurrence and Harris Recurrence -- Splitting Construction and Invariant Measures -- Feller and T-kernels -- Part III Irreducible Chains: Advanced Topics -- Rates of Convergence for Atomic Markov Chains -- Geometric Recurrence and Regularity -- Geometric Rates of Convergence -- (f, r)-recurrence and Regularity -- Subgeometric Rates of Convergence -- Uniform and V-geometric Ergodicity by Operator Methods -- Coupling for Irreducible Kernels -- Part IV Selected Topics -- Convergence in the Wasserstein Distance -- Central Limit Theorems -- Spectral Theory -- Concentration Inequalities -- Appendices -- A Notations -- B Topology, Measure, and Probability -- C Weak Convergence -- D Total and V-total Variation Distances -- E Martingales -- F Mixing Coefficients -- G Solutions to Selected Exercises.
520 $a This book covers the classical theory of Markov chains on general state-spaces as well as many recent developments. The theoretical results are illustrated by simple examples, many of which are taken from Markov Chain Monte Carlo methods. The book is self-contained while all the results are carefully and concisely proven. Bibliographical notes are added at the end of each chapter to provide an overview of the literature. Part I lays the foundations of the theory of Markov chain on general state-spaces. Part II covers the basic theory of irreducible Markov chains starting from the definition of small and petite sets, the characterization of recurrence and transience and culminating in the Harris theorem. Most of the results rely on the splitting technique which allows to reduce the theory of irreducible to a Markov chain with an atom. These two parts can serve as a text on Markov chain theory on general state-spaces. Although the choice of topics is quite different from what is usually covered in a classical Markov chain course, where most of the emphasis is put on countable state space, a graduate student should be able to read almost all of these developments without any mathematical background deeper than that needed to study countable state space (very little measure theory is required) Part III deals with advanced topics on the theory of irreducible Markov chains, covering geometric and subgeometric convergence rates. Special attention is given to obtaining computable convergence bounds using Foster-Lyapunov drift conditions and minorization techniques. Part IV presents selected topics on Markov chains, covering mostly hot recent developments. It represents a biased selection of topics, reflecting the authors own research inclinations. This includes quantitative bounds of convergence in Wasserstein distances, spectral theory of Markov operators, central limit theorems for additive functionals and concentration inequalities. Some of the results in Parts III and IV appear for the first time in book form and some are original.
650 0 $a Markov processes. $3 527825
650 1 4 $a Probability Theory and Stochastic Processes. $3 593945
700 1 $a Douc, Randal. $3 1211836
710 2 $a SpringerLink (Online service) $3 593884
773 0 $t Springer eBooks
830 0 $a Springer series in operations research and financial engineering. $3 884189
856 4 0 $u https://doi.org/10.1007/978-3-319-97704-1
950 $a Mathematics and Statistics (Springer-11649)