Convolution comparison measures

We give a precise functional comparison between classical and free convolutions. If $μ$ and $ν$ are compactly supported probability measures, we show that the expectation of $f$ over the classical convolution $μ* ν$ is at least the expectation of $f$ over the free convolution $μ\boxplus ν$, 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.

💡 Research Summary

The paper establishes a precise functional ordering between the classical convolution μ * ν and the free convolution μ ⊞ ν of compactly supported probability measures on the real line. The central result (Theorem 1.3) states that for any four‑times continuously differentiable function f with a non‑negative fourth derivative everywhere (i.e. f⁽⁴⁾(x) ≥ 0 for all x), the expectation under the classical convolution dominates the expectation under the free convolution: \