The Annals of Probability

An inequality for the distance between densities of free convolutions

V. Kargin

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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

Subjects
Primary: 46L54: Free probability and free operator algebras 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Keywords
Free probability free convolution convergence of measures

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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