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November 2004 Elementary bounds on Poincaré and log-Sobolev constants for decomposable Markov chains
Mark Jerrum, Jung-Bae Son, Prasad Tetali, Eric Vigoda
Ann. Appl. Probab. 14(4): 1741-1765 (November 2004). DOI: 10.1214/105051604000000639

Abstract

We consider finite-state Markov chains that can be naturally decomposed into smaller “projection” and “restriction” chains. Possibly this decomposition will be inductive, in that the restriction chains will be smaller copies of the initial chain. We provide expressions for Poincaré (resp. log-Sobolev) constants of the initial Markov chain in terms of Poincaré (resp. log-Sobolev) constants of the projection and restriction chains, together with further a parameter. In the case of the Poincaré constant, our bound is always at least as good as existing ones and, depending on the value of the extra parameter, may be much better. There appears to be no previously published decomposition result for the log-Sobolev constant. Our proofs are elementary and self-contained.

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Mark Jerrum. Jung-Bae Son. Prasad Tetali. Eric Vigoda. "Elementary bounds on Poincaré and log-Sobolev constants for decomposable Markov chains." Ann. Appl. Probab. 14 (4) 1741 - 1765, November 2004. https://doi.org/10.1214/105051604000000639

Information

Published: November 2004
First available in Project Euclid: 5 November 2004

zbMATH: 1067.60065
MathSciNet: MR2099650
Digital Object Identifier: 10.1214/105051604000000639

Subjects:
Primary: 60J10 , 68W20

Keywords: Decomposition of Markov chains , logarithmic Sobolev inequalities , mixing time of Markov chains , Poincaré inequalities , spectral gap

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.14 • No. 4 • November 2004
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