Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-08T19:29:59.001Z Has data issue: false hasContentIssue false

Regular Perturbation of V-Geometrically Ergodic Markov Chains

Published online by Cambridge University Press:  30 January 2018

Déborah Ferré*
Affiliation:
Institut National des Sciences Appliquées
Loïc Hervé*
Affiliation:
Institut National des Sciences Appliquées
James Ledoux*
Affiliation:
Institut National des Sciences Appliquées
*
Postal address: Mathematical Research Institute of Rennes, Institut National des Sciences Appliquées, 20 Avenue des Buttes de Coesmes, CS 70 839, 35708 Rennes cedex 7, France.
Postal address: Mathematical Research Institute of Rennes, Institut National des Sciences Appliquées, 20 Avenue des Buttes de Coesmes, CS 70 839, 35708 Rennes cedex 7, France.
Postal address: Mathematical Research Institute of Rennes, Institut National des Sciences Appliquées, 20 Avenue des Buttes de Coesmes, CS 70 839, 35708 Rennes cedex 7, France.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, new conditions for the stability of V-geometrically ergodic Markov chains are introduced. The results are based on an extension of the standard perturbation theory formulated by Keller and Liverani. The continuity and higher regularity properties are investigated. As an illustration, an asymptotic expansion of the invariant probability measure for an autoregressive model with independent and identically distributed noises (with a nonstandard probability density function) is obtained.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2013 

References

Altman, E., Avrachenkov, K. E. and Núñez-Queija, R. (2004). Perturbation analysis for denumerable Markov chains with application to queueing models. Adv. Appl. Prob. 36, 839853.Google Scholar
Baladi, V. (2000). Positive Transfer Operators and Decay of Correlations (Adv. Ser. Nonlinear Dynamics 16). World Scientific, River Edge, NJ.Google Scholar
Breyer, L., Roberts, G. O. and Rosenthal, J. S. (2001). A note on geometric ergodicity and floating-point roundoff error. Statist. Prob. Lett. 53, 123127.Google Scholar
Ferré, D. (2011). A parametric first-order edgeworth expansion for markov additive functionals. Application to M-estimations. Preprint. Available at http://hal.archives-ouvertes.fr/hal-00668894.Google Scholar
Gouëzel, S. and Liverani, C. (2006). Banach spaces adapted to Anosov systems. Ergodic Theory Dynam. Systems 26, 189217.Google Scholar
Guibourg, D., Hervé, L. and Ledoux, J. (2012). Quasi-compactness of Markov kernels on weighted-supremum spaces and geometrical ergodicity. Preprint. Available at http://arxiv.org/abs/1110.3240v5.Google Scholar
Heidergott, B. and Hordijk, A. (2003). Taylor series expansions for stationary Markov chains. Adv. Appl. Prob. 35, 10461070.Google Scholar
Hennion, H. (1993). Sur un théorème spectral et son application aux noyaux lipchitziens. Proc. Amer. Math. Soc. 118, 627634.Google Scholar
Hervé, L. and Pène, F. (2010). The Nagaev–Guivarc'h method via the Keller–Liverani theorem. Bull. Soc. Math. France 138, 415489.Google Scholar
Kartashov, N. V. (1981). Strongly stable Markov chains. In Problems of Stability of Stochastic Models (Panevezhis, 1980), Vsesoyuz. Nauch.-Issled. Inst. Sistem. Issled., Moscow, pp. 5459.Google Scholar
Kartashov, N. V. (1996). Strong Stable Markov Chains. VSP, Utrecht.Google Scholar
Keller, G. (1982). Stochastic stability in some chaotic dynamical systems. Monatsh. Math. 94, 313333.Google Scholar
Keller, G. and Liverani, C. (1999). Stability of the spectrum for transfer operators. Ann. Scuola Norm. Sup. Pisa Classe Sci. 28, 141152.Google Scholar
Liverani, C. (2004). Invariant measure and their properties. a functional analytic point of view. In Dynamical Systems, Part II, Nuove Pubblicazioni della Classe di Scienze, Scuola Normale Superiore.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1994). Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Prob. 4, 9811011.Google Scholar
Mouhoubi, Z. and A{ı¨ssani, D.} (2010). New perturbation bounds for denumerable Markov chains. Linear Algebra Appl. 432, 16271649.Google Scholar
Roberts, G. O., Rosenthal, J. S. and Schwartz, P. O. (1998). Convergence properties of perturbed Markov chains. J. Appl. Prob. 35, 111.Google Scholar
Shardlow, T. and Stuart, A. M. (2000). A perturbation theory for ergodic Markov chains and application to numerical approximations. SIAM J. Numer. Analysis 37, 11201137.CrossRefGoogle Scholar
Wu, L. (2004). Essential spectral radius for Markov semigroups. I. Discrete time case. Prob. Theory Relat. Fields 128, 255321.Google Scholar