On Measure Concentration of Vector-Valued Maps

Abstract

In this work, we study concentration properties for vector valued maps. In particular, we describe inequalities which capture the exact dimensional behavior of Lipschitz maps with values in R k . To this task, we study in par-ticular a domination principle for projections which might be of independent interest. We further compare our conclusions to earlier results by Pinelis in the Gaussian case, and discuss extensions to the infinite dimensional setting. Notation: In what follows, whenever we deal with R k , we endow it with the standard Euclidean structure with scalar product · and norm · ·. By γ n , we denote the standard N (0, Id n) Gaussian measure on R n with density dγ n /dx = (2π) −n/2 e −−x 2 /2 . Let g, g 1 , g 2 . . . be independent real N (0, 1) random variables, so that G n = (g 1 , . . . , g n) is an R n -valued normal random vector with distribution γ n . For t ∈ R, let T (t) = γ 1 ([t, ∞)) = P(g ≥ t). Obviously, T (t) = 1 − Φ(t), where Φ is the standard normal distribution function but using the function T will be more convenient in our computations. Let θ be a random vector uniformly distributed on the unit sphere S k−1 ⊆ R k , independent of g, g 1 , g 2 . . .. For the sake of brevity, we denote throughout this work by C, C 1 , C 2 . . . different positive universal constants (i.e. numerical constants which do not depend on n, k or any other parameter). With little effort some more explicit numerical bounds can be deduced from the proofs.