Advertisement

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

  • Loic HervéEmail author
  • James Ledoux
Article
  • 4 Downloads

Abstract

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.

Keywords

Markov chain Rate of convergence Autoregressive models 

Mathematics Subject Classification (2010)

60J05 60J22 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

References

  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

Copyright information

© 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

Personalised recommendations