Small ball probabilities, maximum density and rearrangements

TL;DR

This paper demonstrates that among sums of independent random variables with bounded densities, the probability of the sum lying in a ball is maximized when the variables are uniformly distributed on balls, extending Rogozin's result to higher dimensions.

Contribution

It generalizes Rogozin's maximum density result from the real line to multi-dimensional spaces for sums of independent variables with bounded densities.

Findings

Maximum probability in a ball achieved by uniform distributions on balls

Generalization of Rogozin's result to higher dimensions

Provides a new extremal property for sums of bounded density variables

Abstract

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 balls. This in turn generalizes a result by Rogozin on the maximum density of such sums on the line.