State-Discretization of V-Geometrically Ergodic Markov Chains and Convergence to the Stationary Distribution

  • Loic HervéEmail author
  • James Ledoux


Let \((X_{n})_{n \in \mathbb {N}}\) be a V -geometrically ergodic Markov chain on a measurable space \(\mathbb {X}\) with invariant probability distribution π. In this paper, we propose a discretization scheme providing a computable sequence \((\widehat \pi _{k})_{k\ge 1}\) of probability measures which approximates π as k growths to infinity. The probability measure \(\widehat \pi _{k}\) is computed from the invariant probability distribution of a finite Markov chain. The convergence rate in total variation of \((\widehat \pi _{k})_{k\ge 1}\) to π is given. As a result, the specific case of first order autoregressive processes with linear and non-linear errors is studied. Finally, illustrations of the procedure for such autoregressive processes are provided, in particular when no explicit formula for π is known.


Markov chain Rate of convergence Autoregressive models 

Mathematics Subject Classification (2010)

60J05 60J22 


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  1. Anděl J, Hrach K (2000) On calculation of stationary density of autoregressive processes. Kybernetika (Prague) 36(3):311–319MathSciNetzbMATHGoogle Scholar
  2. Anděl J, Netuka I, Ranocha P (2007) Methods for calculating stationary distribution in linear models of time series. Statistics 41(4):279–287MathSciNetCrossRefGoogle Scholar
  3. Anděl J, Ranocha P (2005) Stationary distribution of absolute autoregression. Kybernetika (Prague) 41(6):735–742MathSciNetzbMATHGoogle Scholar
  4. Borkovec M, Klüppelberg C. (2001) The tail of the stationary distribution of an autoregressive process with ARCH(1) errors. Ann. Appl. Probab. 11(4):1220–1241MathSciNetCrossRefGoogle Scholar
  5. Chan KS, Tong H (1986) A note on certain integral equations associated with nonlinear time series analysis. Probab. Theory Relat. Fields 73(1):153–158MathSciNetCrossRefGoogle Scholar
  6. De Doná J.A., Goodwin GC, Middleton RH, Raeburn I (2000) Convergence of eigenvalues in state-discretization of linear stochastic systems. SIAM J. Matrix Anal. Appl. 21(4):1102–1111MathSciNetCrossRefGoogle Scholar
  7. Ferré D., Hervé L., Ledoux J (2013) Regular perturbation of V-geometrically ergodic Markov chains. J. Appl. Probab. 50:184–194MathSciNetCrossRefGoogle Scholar
  8. Haiman G (1998) Upper and lower bounds for the tail of the invariant distribution of some AR(1) processes. In: Asymptotic methods in probability and statistics (Ottawa, ON, 1997), Amsterdam, pp 723–730Google Scholar
  9. Hervé L., Ledoux J (2014) Approximating Markov chains and V-geometric ergodicity via weak perturbation theory. Stoch. Process. Appl. 124:613–638MathSciNetCrossRefGoogle Scholar
  10. Keller G (1982) Stochastic stability in some chaotic dynamical systems. Monatsh. Math. 94(4):313–333MathSciNetCrossRefGoogle Scholar
  11. Keller G, Liverani C (1999) Stability of the spectrum for transfer operators. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze Sér. 4 28:141–152MathSciNetzbMATHGoogle Scholar
  12. Loges W (2004) The stationary marginal distribution of a threshold AR(1) process. J. Time Ser. Anal. 25(1):103–125MathSciNetCrossRefGoogle Scholar
  13. Meyn SP, Tweedie RL (1993) Markov chains and stochastic stability. Springer-Verlag London Ltd., LondonCrossRefGoogle Scholar
  14. Rudolf D, Schweizer N (2018) Perturbation theory for Markov chains via Wasserstein distance. Bernoulli 24(4A):2610–2639MathSciNetCrossRefGoogle Scholar
  15. Shardlow T, Stuart AM (2000) A perturbation theory for ergodic Markov chains and application to numerical approximations. SIAM J. Numer. Anal. 37:1120–1137MathSciNetCrossRefGoogle Scholar
  16. Truquet L (2017) A perturbation analysis of some Markov chains models with time-varying parameters. arXiv:1706.03214

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Univ Rennes, INSA Rennes, CNRS, IRMAR-UMR 6625Rennes Cedex 7France

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