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001451490 001__ 1451490
001451490 003__ OCoLC
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001451490 019__ $$a1351733489
001451490 020__ $$a9783031118227
001451490 020__ $$a3031118227
001451490 020__ $$z3031118219
001451490 020__ $$z9783031118210
001451490 0247_ $$a10.1007/978-3-031-11822-7$$2doi
001451490 035__ $$aSP(OCoLC)1351746689
001451490 040__ $$aEBLCP$$beng$$erda$$cEBLCP$$dYDX$$dEBLCP$$dGW5XE$$dOCLCF$$dOCLCQ
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001451490 050_4 $$aQA274.7
001451490 08204 $$a519.2/33$$223/eng/20221209
001451490 1001_ $$aBenaïm, Michel,$$eauthor.
001451490 24510 $$aMarkov chains on metric spaces :$$ba short course /$$cMichel Benaïm, Tobias Hurth.
001451490 264_1 $$aCham :$$bSpringer,$$c[2022]
001451490 264_4 $$c©2022
001451490 300__ $$a1 online resource (xv, 197 pages) :$$billustrations.
001451490 336__ $$atext$$btxt$$2rdacontent
001451490 337__ $$acomputer$$bc$$2rdamedia
001451490 338__ $$aonline resource$$bcr$$2rdacarrier
001451490 4901_ $$aUniversitext
001451490 500__ $$aDescription based upon print version of record.
001451490 504__ $$aIncludes bibliographical references and index.
001451490 5050_ $$aIntro -- Preface -- Contents -- Preliminaries -- 1 Markov Chains -- 1.1 Markov Kernels -- 1.2 Markov Chains -- 1.3 The Canonical Chain -- 1.4 Markov and Strong Markov Properties -- 1.5 Continuous Time: Markov Processes -- 2 Countable Markov Chains -- 2.1 Recurrence and Transience -- 2.1.1 Positive Recurrence -- 2.1.2 Null Recurrence -- 2.2 Subsets of Recurrent Sets -- 2.3 Recurrence and Lyapunov Functions -- 2.4 Aperiodic Chains -- 2.5 The Convergence Theorem -- 2.6 Application to Renewal Theory -- 2.6.1 Coupling of Renewal Processes -- 2.7 Convergence Rates for Positive Recurrent Chains
001451490 5058_ $$aNotes -- 3 Random Dynamical Systems -- 3.1 General Definitions -- 3.2 Representation of Markov Chains by RDS -- Notes -- 4 Invariant and Ergodic Probability Measures -- 4.1 Weak Convergence of Probability Measures -- 4.1.1 Tightness and Prohorov's Theorem -- A Tightness Criterion -- 4.2 Invariant Measures -- 4.2.1 Tightness Criteria for Empirical Occupation Measures -- 4.3 Excessive Measures -- 4.4 Ergodic Measures -- 4.5 Unique Ergodicity -- 4.5.1 Unique Ergodicity of Random Contractions -- 4.6 Classical Results from Ergodic Theory -- 4.6.1 Poincaré, Birkhoff, and Ergodic Decomposition Theorems
001451490 5058_ $$a6.1.1 Continuous Time: Doeblin Points for Markov Processes -- 6.2 Random Dynamical Systems -- 6.3 Random Switching Between Vector Fields -- 6.3.1 The Weak Bracket Condition -- 6.4 Piecewise Deterministic Markov Processes -- 6.4.1 Invariant Measures -- 6.4.2 The Strong Bracket Condition -- 6.5 Stochastic Differential Equations -- 6.5.1 Accessibility -- 6.5.2 Hörmander Conditions -- Notes -- 7 Harris and Positive Recurrence -- 7.1 Stability and Positive Recurrence -- 7.2 Harris Recurrence -- 7.2.1 Petite Sets and Harris Recurrence -- 7.3 Recurrence Criteria and Lyapunov Functions
001451490 5058_ $$a7.4 Subsets of Recurrent Sets -- 7.5 Petite Sets and Positive Recurrence -- 7.6 Positive Recurrence for Feller Chains -- 7.6.1 Application to PDMPs -- 7.6.2 Application to SDEs -- 8 Harris Ergodic Theorem -- 8.1 Total Variation Distance -- 8.1.1 Coupling -- 8.2 Harris Convergence Theorems -- 8.2.1 Geometric Convergence -- Aperiodic Small Sets -- 8.2.2 Continuous Time: Exponential Convergence -- 8.2.3 Coupling, Splitting, and Polynomial Convergence -- 8.3 Convergence in Wasserstein Distance -- A Monotone Class and Martingales -- A.1 Monotone Class Theorem -- A.2 Conditional Expectation
001451490 506__ $$aAccess limited to authorized users.
001451490 520__ $$aThis book gives an introduction to discrete-time Markov chains which evolve on a separable metric space. The focus is on the ergodic properties of such chains, i.e., on their long-term statistical behaviour. Among the main topics are existence and uniqueness of invariant probability measures, irreducibility, recurrence, regularizing properties for Markov kernels, and convergence to equilibrium. These concepts are investigated with tools such as Lyapunov functions, petite and small sets, Doeblin and accessible points, coupling, as well as key notions from classical ergodic theory. The theory is illustrated through several recurring classes of examples, e.g., random contractions, randomly switched vector fields, and stochastic differential equations, the latter providing a bridge to continuous-time Markov processes. The book can serve as the core for a semester- or year-long graduate course in probability theory with an emphasis on Markov chains or random dynamics. Some of the material is also well suited for an ergodic theory course. Readers should have taken an introductory course on probability theory, based on measure theory. While there is a chapter devoted to chains on a countable state space, a certain familiarity with Markov chains on a finite state space is also recommended.
001451490 650_0 $$aMarkov processes.
001451490 650_0 $$aMetric spaces.
001451490 655_0 $$aElectronic books.
001451490 7001_ $$aHurth, Tobias,$$eauthor
001451490 77608 $$iPrint version:$$aBenaïm, Michel$$tMarkov Chains on Metric Spaces$$dCham : Springer International Publishing AG,c2022$$z9783031118210
001451490 830_0 $$aUniversitext.
001451490 852__ $$bebk
001451490 85640 $$3Springer Nature$$uhttps://univsouthin.idm.oclc.org/login?url=https://link.springer.com/10.1007/978-3-031-11822-7$$zOnline Access$$91397441.1
001451490 909CO $$ooai:library.usi.edu:1451490$$pGLOBAL_SET
001451490 980__ $$aBIB
001451490 980__ $$aEBOOK
001451490 982__ $$aEbook
001451490 983__ $$aOnline
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