Ergodic Theory of Markov and Related Processes

20 Aug 2007 21:31

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Yet Another Inadequate Placeholder of references.

See: ergodic theory; interacting particle systems; Markov and hidden Markov models; Monte Carlo; non-equilibrium statistical mechanics

Recommended:
• J. Doob, Stochastic Processes [A good source for classical results]
• Shaul R. Foguel, The Ergodic Theory of Markov Processes
• Andrzej Lasota and Michael C. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics
• Murray Rosenblatt, Markov Processes: Structure and Asymptotic Behavior

To read:
• Peter H. Baxendale, "Renewal theory and computable convergence rates for geometrically ergodic Markov chains", Annals of Applied Probability 15 (2005): 700--738 = math.PR/0503515
• Itai Benjamini, Krzysztof Burdzy, Zhen-Qing Chen, "Shy couplings", math.PR/0509458 ["We say that a coupling is ``shy'' if the processes never come closer than some (random) strictly positive distance from each other."]
• Gordon Blower and Francois Bolley, "Concentration inequalities on product spaces with applications to Markov processes", math.PR/0505536
• Anne-Severine Boudou, Pietro Caputo, Paolo Dai Pra and Gustavo Posta, "Spectral gap estimates for interacting particle systems via a Bakry & Emery-type approach", math.PR/0505533 ["We develop a general technique, based on the Bakry-Emery approach, to estimate spectral gaps of a class of Markov operators. We apply this technique to various interacting particle systems."]
• Anton Bovier, Michael Eckhoff, Veronique Gayrard and Markus Klein, "Metastability and Small Eigenvalues in Markov Chains," cond-mat/0007343
• Patrick Cattiaux and Arnaud Guillin, "Trends to Equilibrium in Total Variation Distance", math.PR/0703451
• Mu-Fa Chen
  • Eigenvalues, Inequalities, and Ergodic Theory [Blurb]
  • "Ergodic convergence rates of Markov processes--eigenvalues, inequalities and ergodic theory", math.PR/0304367
• Xia Chen, Limit Theorems for Functionals of Ergodic Markov Chains with General State Space [blurb]
• Persi Diaconis and David Freedman, "On Markov Chains with Continuous State Space", UCB Statistics Tech. Report #501 ["In this expository paper, we prove the following theorem... Suppose a discrete-time Markov chain is aperiodic, irreducible, and there is a stationary probability distribution. Then from almost all starting points the distribution of the chain at time n converges in norm to the stationary distribution. This known theorem is a special case of more general results due to Doeblin, and the paper conclues with a brief review of the literature." PS.Z]
• G. B. DiMasi and L. Stettner, "Ergodicity of hidden Markov models", Mathematics of Control, Signals, and Systems 17 (2005): 269--296
• R. Douc, E. Moulines, and Jeffrey S. Rosenthal, "Quantitative bounds on convergence of time-inhomogeneous Markov chains", Annals of Applied Probability 14 (2004): 1643--1665 = math.PR/0403532
• Olle Häggström, "On the central limit theorem for geometrically ergodic Markov chains", Probability Theory and Related Fields 132 (2005): 74--82
• Stefano Isola
  • "On the rate of convergence to equilibrium for countable ergodic Markov chains", math.PR/0308018 ["Using elementary methods, we prove that for a countable Markov chain $P$ of ergodic degree $d > 0$ the rate of convergence towards the stationary distribution is subgeometric of order $n^{-d}$, provided the initial distribution satisfies certain conditions of asymptotic decay. ... Explicit conditions allowing to obtain the actual asymptotics for the rate of convergence are also discussed."]
  • "On systems with finite ergodic degree", math.DS/0308019
• Leonid Kontorovich, "Measure Concentration of Hidden Markov Processes", math.PR/0608064 [I am really not sure why Leo bothered to take my class...]
• Andreas Nordvall Lageras and Orjan Stenflo, "Central limit theorems for contractive Markov chains", Nonlinearity 18 (2005): 1955--1965
• Neal Madras and Dana Randall, "Markov chain decomposition for convergence rate analysis", Annals of Applied Probability 12 (2002): 581--606
• Sean P. Meyn and Richard L. Tweedie, Markov Chains and Stochastic Stability [Full text free online, courtesy of Prof. Meyn.]
• F. Rassoul-Agha and T. Seppalainen, "An Almost Sure Invariance Principle for Additive Functionals of Markov Chains", math.PR/0411603
• Sunder Sethuraman and S. R. S. Varadhan, "A martingale proof of Dobrushin's theorem for non-homogeneous Markov chains", math.PR/0404231 [As you know, Bob, Dobrushin's theorem is a central limit theorem for Markov chains]
• Wojciech Slomczynski, Dynamical Entropy, Markov Operators, and Iterated Function Systems [Many thanks to Dr. Slomczynski for sending a copy of his work]
• Ivan Werner [The sequence of papers on contractive Markov systems look very important, but I keep not finding the time to read them...]
  • "Contractive Markov Systems", Journal of the London Mathematical Society 71 (2005): 236--258
  • "Contractive Markov systems II", math.PR/0503633
  • "Fundamental Markov systems", math.PR/0509120
  • "The generalized Markov measure as an equilibrium state", math.DS/0503644 = Nonlinearity 18 (2005): 2261--2274
  • "Kolmogorov-Sinai entropy of a generalized Markov shift", math.DS/0502389
  • "On coding by Feller contractive Markov systems", math.DS/0506476
  • "A necessary condition for the uniqueness of the stationary state of a Markov system", math.PR/0508054