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

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.