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August 2017 The $L^{2}$-cutoffs for reversible Markov chains
Guan-Yu Chen, Jui-Ming Hsu, Yuan-Chung Sheu
Ann. Appl. Probab. 27(4): 2305-2341 (August 2017). DOI: 10.1214/16-AAP1260

Abstract

In this article, we considers reversible Markov chains of which $L^{2}$-distances can be expressed in terms of Laplace transforms. The cutoff of Laplace transforms was first discussed by Chen and Saloff-Coste in [J. Funct. Anal. 258 (2010) 2246–2315], while we provide here a completely different pathway to analyze the $L^{2}$-distance. Consequently, we obtain several considerably simplified criteria and this allows us to proceed advanced theoretical studies, including the comparison of cutoffs between discrete time lazy chains and continuous time chains. For an illustration, we consider product chains, a rather complicated model which could be involved to analyze using the method in [J. Funct. Anal. 258 (2010) 2246–2315], and derive the equivalence of their $L^{2}$-cutoffs.

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Guan-Yu Chen. Jui-Ming Hsu. Yuan-Chung Sheu. "The $L^{2}$-cutoffs for reversible Markov chains." Ann. Appl. Probab. 27 (4) 2305 - 2341, August 2017. https://doi.org/10.1214/16-AAP1260

Information

Received: 1 June 2016; Revised: 1 November 2016; Published: August 2017
First available in Project Euclid: 30 August 2017

zbMATH: 1374.60130
MathSciNet: MR3693527
Digital Object Identifier: 10.1214/16-AAP1260

Subjects:
Primary: 60J10 , 60J27

Keywords: Cutoff phenomenon , Product chains

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.27 • No. 4 • August 2017
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