Small Ball Probabilities for Linear Images of High-Dimensional Distributions

We study concentration properties of random vectors of the form |$AX$|⁠, where |X = (X₁,… , 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\|HS| is at most |(Cp)0.9 r(A)|⁠, where |$r(A)$| denotes the stable rank of |A| and |C| is an absolute constant.