The Annals of Applied Probability

Elementary bounds on Poincaré and log-Sobolev constants for decomposable Markov chains

Mark Jerrum, Jung-Bae Son, Prasad Tetali, and Eric Vigoda

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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.

Article information

Source
Ann. Appl. Probab. Volume 14, Number 4 (2004), 1741-1765.

Dates
First available in Project Euclid: 5 November 2004

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1099674076

Digital Object Identifier
doi:10.1214/105051604000000639

Mathematical Reviews number (MathSciNet)
MR2099650

Zentralblatt MATH identifier
02148331

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 68W20: Randomized algorithms

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

Citation

Jerrum, Mark; Son, Jung-Bae; Tetali, Prasad; Vigoda, Eric. Elementary bounds on Poincaré and log-Sobolev constants for decomposable Markov chains. The Annals of Applied Probability 14 (2004), no. 4, 1741--1765. doi:10.1214/105051604000000639. http://projecteuclid.org/euclid.aoap/1099674076.


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