AN INEQUALITY FOR THE DISTANCE BETWEEN DENSITIES OF FREE CONVOLUTIONS

• 成果类型: Article
• 署名作者: Kargin, V.
• 署名单位: University of Cambridge
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP756
• 发表日期: 2013
• 页码: 3241-3260
• 关键词: free probability-theory; random-matrix theory; information measure; random-variables; limit; entropy; THEOREM; analogs; LAWS

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.