Concentration and deviation inequalities in infinite dimensions via covariance representations

Abstract

Concentration and deviation inequalities are obtained for functionals on Wiener space, Poisson space or more generally for normal martingales and binomial processes. The method used here is based on covariance identities obtained via the chaotic representation property, and provides an alternative to the use of logarithmic Sobolev inequalities. It enables the recovery of known concentration and deviation inequalities on the Wiener and Poisson space (including those given by sharp logarithmic Sobolev inequalities), and extends results available in the discrete case, i.e. on the infinite cube $\{-1,1\}^\infty$.