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September 2016 >Computable bounds of an 𝓁²-spectral gap for discrete Markov chains with band transition matrices
Loïc Hervé, James Ledoux
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J. Appl. Probab. 53(3): 946-952 (September 2016).

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.

Citation

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Loïc Hervé. James Ledoux. ">Computable bounds of an 𝓁²-spectral gap for discrete Markov chains with band transition matrices." J. Appl. Probab. 53 (3) 946 - 952, September 2016.

Information

Published: September 2016
First available in Project Euclid: 13 October 2016

zbMATH: 1351.60092
MathSciNet: MR3570107

Subjects:
Primary: 47B07 , 60J10
Secondary: 60F99 , 60J80

Keywords: essential spectral radius , 𝑉-geometric ergodicity

Rights: Copyright © 2016 Applied Probability Trust

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Vol.53 • No. 3 • September 2016
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