Small ball probabilities, maximum density and rearrangements

T. Juškevičius, J. D. Lee

Published 2015-03-31Version 1

We prove that the probability that a sum of independent random variables in $\mathbb{R}^d$ with bounded densities lies in a ball is maximized by taking uniform distributions on spheres. This in turn generalizes a result by Rogozin on the maximum density of such sums on the line.