Regularity and Convergence Properties of Finite Free Convolutions

Finite free convolutions, ${⊞}_d$ and ${⊠}_d$, are binary operations on polynomials of degree $d$ that are central to finite free probability, a developing field at the intersection of free probability and the geometry of polynomials. Motivated by established regularities in free probability, this paper investigates analogous regularities for finite free convolutions. Key findings include triangle inequalities for these convolutions and necessary and sufficient conditions regarding atoms of probability measures. Applications of these results include proving the weak convergence of ${⊞}_d$ and ${⊠}_d$ to their infinite counterparts $⊞$ and $⊠$ as $d\to \mathrm{\infty }$, without compactness assumptions. Furthermore, this weak convergence is strengthened to convergence in Kolmogorov distance.