Small Ball Probabilities for Linear Images of High-Dimensional Distributions

Abstract

We study concentration properties of random vectors of the form AX, where X = (X1,\ldots, Xn) has independent coordinates and A is a given matrix. We show that the distribution of AX is well spread in space whenever the distributions of Xi are well spread on the line. Specifically, assume that the probability that Xi falls in any given interval of length t is at most p. Then the probability that AX falls in any given ball of radius t \|A\|{\mathrm {HS}} is at most (Cp){0.9 r(A)}, where r(A) denotes the stable rank of A and C is an absolute constant.