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On The Law Of Large Numbers For (Geometrically) Ergodic Markov Chains

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  • Jensen, Søren Tolver
  • Rahbek, Anders

Abstract

For use in asymptotic analysis of nonlinear time series models, we show that with (Xt,t ≥ 0) a (geometrically) ergodic Markov chain, the general version of the strong law of large numbers applies. That is, the average converges almost surely to the expectation of φ(Xt,Xt+1,…) irrespective of the choice of initial distribution of, or value of, X0. In the existing literature, the less general form has been studied.We thank Paolo Paruolo (the co-editor) and the referee for valuable comments. Also we thank the Danish Social Sciences Research Council (grant 2114-04-0001) for financial support.

Suggested Citation

  • Jensen, Søren Tolver & Rahbek, Anders, 2007. "On The Law Of Large Numbers For (Geometrically) Ergodic Markov Chains," Econometric Theory, Cambridge University Press, vol. 23(4), pages 761-766, August.
    • Handle: RePEc:cup:etheor:v:23:y:2007:i:04:p:761-766_07
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    Cited by:

    1. Fokianos, Konstantinos & Rahbek, Anders & Tjøstheim, Dag, 2009. "Poisson Autoregression," Journal of the American Statistical Association, American Statistical Association, vol. 104(488), pages 1430-1439.
    2. Frédérique Bec & Anders Rahbek & Neil Shephard, 2008. "The ACR Model: A Multivariate Dynamic Mixture Autoregression," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 70(5), pages 583-618, October.
    3. Cheikh Toure & Anne Auger & Nikolaus Hansen, 2023. "Global linear convergence of evolution strategies with recombination on scaling-invariant functions," Journal of Global Optimization, Springer, vol. 86(1), pages 163-203, May.
    4. Peter Boswijk, H. & van der Weide, Roy, 2011. "Method of moments estimation of GO-GARCH models," Journal of Econometrics, Elsevier, vol. 163(1), pages 118-126, July.
    5. Kristensen, Dennis & Rahbek, Anders, 2010. "Likelihood-based inference for cointegration with nonlinear error-correction," Journal of Econometrics, Elsevier, vol. 158(1), pages 78-94, September.
    6. Cavaliere, Giuseppe & Rahbek, Anders & Taylor, A.M. Robert, 2010. "Cointegration Rank Testing Under Conditional Heteroskedasticity," Econometric Theory, Cambridge University Press, vol. 26(6), pages 1719-1760, December.
    7. Konstantinos Fokianos & Dag Tjøstheim, 2012. "Nonlinear Poisson autoregression," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(6), pages 1205-1225, December.
    8. Nielsen, Heino Bohn & Rahbek, Anders, 2014. "Unit root vector autoregression with volatility induced stationarity," Journal of Empirical Finance, Elsevier, vol. 29(C), pages 144-167.
    9. Christian M. Dahl & Emma M. Iglesias, 2021. "Asymptotic normality of the MLE in the level-effect ARCH model," Statistical Papers, Springer, vol. 62(1), pages 117-135, February.
    10. Vurukonda Sathish & Siuli Mukhopadhyay & Rashmi Tiwari, 2022. "Autoregressive and moving average models for zero‐inflated count time series," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 76(2), pages 190-218, May.
    11. Theis Lange, 2009. "First and second order non-linear cointegration models," CREATES Research Papers 2009-04, Department of Economics and Business Economics, Aarhus University.
    12. Hanan Elsaied & Roland Fried, 2021. "On robust estimation of negative binomial INARCH models," METRON, Springer;Sapienza Università di Roma, vol. 79(2), pages 137-158, August.
    13. Line Elvstrøm Ekner & Emil Nejstgaard, 2013. "Parameter Identification in the Logistic STAR Model," Discussion Papers 13-07, University of Copenhagen. Department of Economics.
    14. Giuseppe Cavaliere & Thomas Mikosch & Anders Rahbek & Frederik Vilandt, 2023. "Asymptotics for the Generalized Autoregressive Conditional Duration Model," Papers 2307.01779, arXiv.org.
    15. Kristensen Dennis & Rahbek Anders, 2009. "Asymptotics of the QMLE for Non-Linear ARCH Models," Journal of Time Series Econometrics, De Gruyter, vol. 1(1), pages 1-38, April.

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