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

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

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.
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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 r ess(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=−N N lim supi→+∞(P(i,i+m)P *(i+m,i)1∕2<1. Moreover, r ess(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 

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