Spectral gap and convex concentration inequalities for birth-death processes

Abstract

In this paper, we consider a birth-death process with generator ℒ and reversible invariant probability π. Given an increasing function ρ and the associated Lipschitz norm ∥ · ∥Lip(ρ), we find an explicit formula for ∥(- ℒ)-1∥Lip(ρ). As a typical application, with spectral theory, we revisit one variational formula of M. F. Chen for the spectral gap of ℒ in L2(π). Moreover, by Lyons-Zheng's forward-backward martingale decomposition theorem, we get convex concentration inequalities for additive functionals of birth-death processes. © 2009 Association des Publications de l'Institut Henri Poincaré.