Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-24T22:12:33.149Z Has data issue: false hasContentIssue false
  • English
  • Français

Computable bounds of an 𝓁²-spectral gap for discrete Markov chains with band transition matrices

Part of: Limit theorems Special classes of linear operators Markov processes

Published online by Cambridge University Press:  24 October 2016

Loï Hervé  and
James Ledoux
Loï Hervé*
Affiliation:
INSA de Rennes
James Ledoux*
Affiliation:
INSA de Rennes
*
* Postal address: INSA de Rennes, IRMAR CNRS-UMR 6625, 20 avenue des Buttes de Coesmes, CS 70 839, 35708 Rennes cedex 7, France.
* Postal address: INSA de Rennes, IRMAR CNRS-UMR 6625, 20 avenue des Buttes de Coesmes, CS 70 839, 35708 Rennes cedex 7, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

We analyse the 𝓁²(𝜋)-convergence rate of irreducible and aperiodic Markov chains with N-band transition probability matrix P and with invariant distribution 𝜋. This analysis is heavily based on two steps. First, the study of the essential spectral radius ress(P|𝓁²(𝜋)) of P|𝓁²(𝜋) derived from Hennion’s quasi-compactness criteria. Second, the connection between the spectral gap property (SG2) of P on 𝓁²(𝜋) and the V-geometric ergodicity of P. Specifically, the (SG2) is shown to hold under the condition α0≔∑m=−NNlim supi→+∞(P(i,i+m)P*(i+m,i)1∕2<1. Moreover, ress(P|𝓁²(𝜋)≤α0. Effective bounds on the convergence rate can be provided from a truncation procedure.

Keywords

V-geometric ergodicityessential spectral radius

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 47B07: Operators defined by compactness properties
Secondary: 60F99: None of the above, but in this section 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 3 , September 2016 , pp. 946 - 952
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Baxendale, P. H. (2005).Renewal theory and computable convergence rates for geometrically ergodic Markov chains.Ann. Appl. Prob. 15,700–738.Google Scholar
[2] Chen, M.-F. (2004).From Markov Chains to Non-Equilibrium Particle Systems, 2nd edn.World Scientific ,River Edge, NJ.Google Scholar
[3] Hennion, H. (1993).Sur un théorème spectral et son application aux noyaux lipchitziens.Proc. Amer. Math. Soc. 118,627–634.Google Scholar
[4] Hervé, L. and Ledoux, J. (2014).Approximating Markov chains and V-geometric ergodicity via weak perturbation theory.Stoch. Process. Appl. 124,613–638.Google Scholar
[5] Hervé, L. and Ledoux, J. (2014).Spectral analysis of Markov kernels and application to the convergence rate of discrete random walks.Adv. Appl. Prob. 46,1036–1058.Google Scholar
[6] Hervé, L. and Ledoux, J. (2015).Additional material on bounds of 𝓁²-spectral gap for discrete Markov chains with band transition matrices. Preprint. Available at http://arxiv.org/abs/1503.02206.Google Scholar
[7] Kontoyiannis, I. and Meyn, S. P. (2012).Geometric ergodicity and the spectral gap of non-reversible Markov chains.Prob. Theory Relat. Fields 154,327–339.Google Scholar
[8] Mao, Y. H. and Song, Y. H. (2013).Spectral gap and convergence rate for discrete-time Markov chains.Acta Math. Sin. (Engl. Ser.) 29,1949–1962.Google Scholar
[9] Roberts, G. O. and Rosenthal, J. S. (1997).Geometric ergodicity and hybrid Markov chains.Electron. Commun. Prob. 2,13–25.CrossRefGoogle Scholar
[10] Rosenblatt, M. (1971).Markov Processes: Structure and Asymptotic Behavior.Springer,New York.CrossRefGoogle Scholar
[11] Stadje, W. and Wübker, A. (2011).Three kinds of geometric convergence for Markov chains and the spectral gap property.Electron. J. Prob. 16,1001–1019.CrossRefGoogle Scholar
[12] Wu, L. (2004).Essential spectral radius for Markov semigroups. I. Discrete time case.Prob. Theory Relat. Fields 128,255–321.Google Scholar
[13] Wübker, W. (2012).Spectral theory for weakly reversible Markov chains.J. Appl. Prob. 49,245–265.CrossRefGoogle Scholar
[14] Yuan, W. K. (2000).Applications of geometric bounds to the convergence rate of Markov chains on ℝ n .Stoch. Process. Appl. 87,1–23.Google Scholar