A measure concentration inequality for contracting markov chains

Abstract

The concentration of measure phenomenon in product spaces means the following: if a subset A of the n'th power of a probability space A' does not have too small a probability then very large probability is concentrated in a small neighborhood of A. The neighborhood is in many cases understood in the sense of Hamming distance, and then measure concentration is known to occur for product probability measures, and also for the distribution of some processes with very fast and uniform decay of memory. Recently Talagrand introduced another notion of neighborhood of sets for which he proved a similar measure concentration inequality which in many cases allows more efficient applications than the one for a Hamming neighborhood. So far this inequality has only been proved for product distributions. The aim of this paper is to give a new proof of Talagrand's inequality, which admits an extension to contracting Markov chains. The proof is based on a new asymmetric notion of distance between probability measures, and bounding this distance by informational divergence.As an application, we analyze the bin packing problem for Markov chains.

Chernoff and Berry–Esséen inequalities for Markov processes

“…In [16] the saddle point method is used to obtain the Edgeworth expansions of the deviation probability. Finally, an approach based on the concentration inequalities developped by [22] and [24] gives deviation inequalities for Markov chains.…”

Transportation-information inequalities for Markov processes

“…The usual transportation inequalities for a given µ ∈ M 1 (X ), introduced by K. Marton [31,32] and M. Talagrand [37], compare the Wasserstein metric W p (ν, µ) with the relative entropy H(ν|µ). The following extension of these inequalities:…”

On Talagrand's deviation inequalities for product measures

“…While it is uncertain whether our approach could recover these abstract principles, the deviation inequalities themselves follow rather easily from it. On the abstract inequalities themselves, let us mention here the recent a l t e rnate approach b y K. Marton (1995aMarton ( ), (1995b and Dembo (1995) (see also Dembo and Zeitouni (1995)) based on information inequalities and coupling in which the concept of entropy a l s o p l a ys a crucial role. Let us also observe that hypercontraction methods were used in Kwapie n and Szulga (1991) to study integrability of norms of sums of independent v ector valued random variables.…”