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Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space

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  • Tweedie, Richard L.
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    Abstract

    Let {Xn} be a [empty set][combining character]-irreducible Markov chain on an arbitrary space. Sufficient conditions are given under which the chain is ergodic or recurrent. These extend known results for chains on a countable state space. In particular, it is shown that if the space is a normed topological space, then under some continuity conditions on the transition probabilities of {Xn} the conditions for ergodicity will be met if there is a compact set K and an [epsilon] > 0 such that E {||Xn+1|| -- ||Xn|| | Xn = x} [less-than-or-equals, slant] -[epsilon] whenever x lies outside K and E{||Xn+1|| | Xn=x} is bounded, x [set membership, variant] K; whilst the conditions for recurrence will be met if there exists a compact K with E {||Xn+1|| - ||Xn|| | Xn = x} [less-than-or-equals, slant] 0 for all x outside K. An application to queueing theory is given.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 3 (1975)
    Issue (Month): 4 (October)
    Pages: 385-403

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    Handle: RePEc:eee:spapps:v:3:y:1975:i:4:p:385-403

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    Keywords: recurrence invariant measures positive recurrence stationary measures ergodicity waiting time Markov chains dependent queues;

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    Cited by:
    1. James Mitchell & George Kapetanios & Yongcheol Shin, 2012. "A Nonlinear Panel Data Model of Cross-Sectional Dependence," Discussion Papers in Economics 12/01, Department of Economics, University of Leicester.
    2. Kapetanios, George, 2006. "Nonlinear autoregressive models and long memory," Economics Letters, Elsevier, vol. 91(3), pages 360-368, June.
    3. Hwang, S.Y. & Basawa, I.V., 2009. "Branching Markov processes and related asymptotics," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1155-1167, July.
    4. Addo, Peter Martey & Billio, Monica & Guégan, Dominique, 2014. "The univariate MT-STAR model and a new linearity and unit root test procedure," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 4-19.
    5. Andy Snell & George Kapetanios & Yongcheol Shin, 2004. "Testing for nonlinear cointegration between stock prices and dividends," Money Macro and Finance (MMF) Research Group Conference 2003 90, Money Macro and Finance Research Group.
    6. Hwang, Sun Y. & Basawa, I. V., 2001. "Nonlinear time series contiguous to AR(1) processes and a related efficient test for linearity," Statistics & Probability Letters, Elsevier, vol. 52(4), pages 381-390, May.
    7. Tweedie, R. L., 2001. "Drift conditions and invariant measures for Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 92(2), pages 345-354, April.
    8. Onur Ozgur & Alberto Bisin, 2011. "Dynamic linear economies with social interactions," Levine's Working Paper Archive 786969000000000036, David K. Levine.
    9. George Kapetanios & Tony Yates, 2014. "Evolving UK and US macroeconomic dynamics through the lens of a model of deterministic structural change," Empirical Economics, Springer, vol. 47(1), pages 305-345, August.
    10. Kapetanios, G. & Tzavalis, E., 2010. "Modeling structural breaks in economic relationships using large shocks," Journal of Economic Dynamics and Control, Elsevier, vol. 34(3), pages 417-436, March.
    11. Zhen, X. & Basawa, I.V., 2009. "Categorical time series models for contingency tables," Statistics & Probability Letters, Elsevier, vol. 79(10), pages 1331-1336, May.
    12. Cline, Daren B. H. & Pu, Huay-min H., 2002. "A note on a simple Markov bilinear stochastic process," Statistics & Probability Letters, Elsevier, vol. 56(3), pages 283-288, February.
    13. George Kapetanios & Yongcheol Shin, 2004. "Unit Root Tests in Three-Regime SETAR Models," ESE Discussion Papers 104, Edinburgh School of Economics, University of Edinburgh.
    14. Baillie, Richard T. & Kapetanios, George, 2008. "Nonlinear models for strongly dependent processes with financial applications," Journal of Econometrics, Elsevier, vol. 147(1), pages 60-71, November.
    15. Sollis, Robert, 2009. "A simple unit root test against asymmetric STAR nonlinearity with an application to real exchange rates in Nordic countries," Economic Modelling, Elsevier, vol. 26(1), pages 118-125, January.
    16. Jaya Krishnakumar & David Neto, 2005. "Partial Cointegration," Research Papers by the Institute of Economics and Econometrics, Geneva School of Economics and Management, University of Geneva 2005.04, Institut d'Economie et Econométrie, Université de Genève, revised Aug 2006.
    17. Kapetanios, George & Shin, Yongcheol & Snell, Andy, 2003. "Testing for a unit root in the nonlinear STAR framework," Journal of Econometrics, Elsevier, vol. 112(2), pages 359-379, February.

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