Tail-sensitive Gaussian asymptotics for marginals of concentrated measures in high dimension

Abstract

If the Euclidean norm is strongly concentrated with respect to a measure, the average distribution of an average marginal of this measure has Gaussian asymptotics that captures tail behaviour. If the marginals of the measure have exponential moments, Gaussian asymptotics for the distribution of the average marginal implies Gaussian asymptotics for the distribution of most individual marginals. We show applications to measures of geometric origin.