An inequality for the distance between densities of free convolutions

Abstract

This paper contributes to the study of the free additive convolution of probability measures. It shows that under some conditions, if measures $\mu_i$ and $\nu_i, i=1,2$, are close to each other in terms of the L\'{e}vy metric and if the free convolution $\mu_1\boxplus\mu_2$ is sufficiently smooth, then $\nu_1\boxplus\nu_2$ is absolutely continuous, and the densities of measures $\nu_1\boxplus\nu_2$ and $\mu_1\boxplus\mu_2$ are close to each other. In particular, convergence in distribution $\mu_1^{(n)}\rightarrow \mu_1,$ $\mu_2^{(n)}\rightarrow\mu_2$ implies that the density of $\mu_1^{(n)}\boxplus\mu_2^{(n)}$ is defined for all sufficiently large $n$ and converges to the density of $\mu_1\boxplus\mu_2$. Some applications are provided, including: (i) a new proof of the local version of the free central limit theorem, and (ii) new local limit theorems for sums of free projections, for sums of $\boxplus$-stable random variables and for eigenvalues of a sum of two $N$-by-$N$ random matrices.