## The Annals of Probability

### An inequality for the distance between densities of free convolutions

V. Kargin

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 41, Number 5 (2013), 3241-3260.

Dates
First available in Project Euclid: 12 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1378991838

Digital Object Identifier
doi:10.1214/12-AOP756

Mathematical Reviews number (MathSciNet)
MR3127881

Zentralblatt MATH identifier
1284.46055

#### Citation

Kargin, V. An inequality for the distance between densities of free convolutions. Ann. Probab. 41 (2013), no. 5, 3241--3260. doi:10.1214/12-AOP756. https://projecteuclid.org/euclid.aop/1378991838

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