Open Access
September 2013 An inequality for the distance between densities of free convolutions
V. Kargin
Ann. Probab. 41(5): 3241-3260 (September 2013). DOI: 10.1214/12-AOP756


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 Lé vy metric and if the free convolution $\mu_{1}\boxplus\mu_{2}$ is sufficiently smooth, then $\nu_{1}\boxplus\nu_{2}$ is absolutely continuous, and the densities of measures $\nu_{1}\boxplus\nu_{2}$ and $\mu_{1}\boxplus\mu_{2}$ are close to each other. In particular, convergence in distribution $\mu_{1}^{(n)}\rightarrow\mu_{1}$, $\mu_{2}^{(n)}\rightarrow\mu_{2}$ implies that the density of $\mu_{1}^{(n)}\boxplus\mu_{2}^{(n)}$ is defined for all sufficiently large $n$ and converges to the density of $\mu_{1}\boxplus\mu_{2}$. 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 $\boxplus$-stable random variables and for eigenvalues of a sum of two $N$-by-$N$ random matrices.


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V. Kargin. "An inequality for the distance between densities of free convolutions." Ann. Probab. 41 (5) 3241 - 3260, September 2013.


Published: September 2013
First available in Project Euclid: 12 September 2013

zbMATH: 1284.46055
MathSciNet: MR3127881
Digital Object Identifier: 10.1214/12-AOP756

Primary: 46L54 , 60B20

Keywords: convergence of measures , Free convolution , Free probability

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 5 • September 2013
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