科研之友
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摘要
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 Levy metric and if the free convolution mu(1) boxed plus mu(2) is sufficiently smooth, then nu(1) boxed plus nu(2) is absolutely continuous, and the densities of measures nu(1) boxed plus nu(2) and mu(1) boxed plus mu(2) are close to each other. In particular, convergence in distribution mu((n))(1) -> mu(1), mu((n))(2) -> mu(2) implies that the density of mu((n))(1) boxed plus mu((n))(2) is defined for all sufficiently large n and converges to the density of in mu(1) boxed plus 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 boxed plus-stable random variables and for eigenvalues of a sum of two N-by-N random matrices.
- 出版日期2013-9
