Convolution comparison measures

TL;DR

This paper establishes a precise comparison between classical and free convolutions of probability measures, showing that under certain conditions on a function's derivatives, the classical convolution expectation exceeds that of the free convolution.

Contribution

It provides the first exact functional comparison between classical and free convolutions, including necessary and sufficient conditions based on the fourth derivative of functions.

Findings

Classical convolution expectation is at least the free convolution expectation for functions with non-negative fourth derivative.

The non-negativity of the fourth derivative is necessary for the comparison.

A new measure, the convolution comparison measure, is introduced and expressed via Hermitian matrices.

Abstract

We give a precise functional comparison between classical and free convolutions. If $\mu$ and $\nu$ are compactly supported probability measures, we show that the expectation of $f$ over the classical convolution $\mu * \nu$ is at least the expectation of $f$ over the free convolution $\mu \boxplus \nu$, as long as the fourth derivative of $f$ is non-negative. Conversely, the non-negativity of the fourth derivative is necessary for such a comparison. This comparison is based on the positivity of a related measure on $\mathbb{R}^{2}$, which we dub the convolution comparison measure. We give an expression for this measure using a curious identity involving Hermitian matrices.