Paper detail

An inequality for the distance between densities of free convolutions

This paper contributes to the study of the free additive convolution of probability measures. It shows that under some conditions, if measures $μ_i$ and $ν_i, i=1,2$, are close to each other in terms of the Lévy metric and if the free convolution $μ_1\boxplusμ_2$ is sufficiently smooth, then $ν_1\boxplusν_2$ is absolutely continuous, and the densities of measures $ν_1\boxplusν_2$ and $μ_1\boxplusμ_2$ are close to each other. In particular, convergence in distribution $μ_1^{(n)}\rightarrow μ_1,$ $μ_2^{(n)}\rightarrowμ_2$ implies that the density of $μ_1^{(n)}\boxplusμ_2^{(n)}$ is defined for all sufficiently large $n$ and converges to the density of $μ_1\boxplusμ_2$. 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.