Resumen de 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 μi and νi, i=1,2, are close to each other in terms of the Lé vy metric and if the free convolution μ1⊞μ2 is sufficiently smooth, then ν1⊞ν2 is absolutely continuous, and the densities of measures ν1⊞ν2 and μ1⊞μ2 are close to each other. In particular, convergence in distribution μ(n)1→μ1, μ(n)2→μ2 implies that the density of μ(n)1⊞μ(n)2 is defined for all sufficiently large n and converges to the density of μ1⊞μ2. 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 ⊞-stable random variables and for eigenvalues of a sum of two N-by-N random matrices.
