Mathematics – Probability
Scientific paper
2011-07-03
Mathematics
Probability
18 pages, 2 figures
Scientific paper
It is shown that under some conditions the distance between densities of free convolutions of two pairs of probability measures is smaller than the maximum of the Levy distance between the corresponding measures in these pairs. In particular, weak convergence of measures \mu_n to \mu, and measures \nu_n to \nu, implies that free convolution of \mu_n and \nu_n has a density for all sufficiently large n, and this density converges to the density of the free convolution of \mu and \nu. Some applications are provided including (i) a new proof of the local version of the free central limit theorem, (ii) a local limit theorem for sums of free projections, and (iii) a local limit theorem for sums of free stable random variables. In addition, a local limit law for eigenvalues of a sum of N-by-N random matrices is proved.
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