#### Vol. 292, No. 2, 2018

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Superconvergence to freely infinitely divisible distributions

### Hari Bercovici, Jiun-Chau Wang and Ping Zhong

Vol. 292 (2018), No. 2, 273–291
##### Abstract

We prove superconvergence results for all freely infinitely divisible distributions. Given a nondegenerate freely infinitely divisible distribution $\nu$, let ${\mu }_{n}$ be a sequence of probability measures and let ${k}_{n}$ be a sequence of integers tending to infinity such that ${\mu }_{n}^{⊞{k}_{n}}$ converges weakly to $\nu$. We show that the density $d{\mu }_{n}^{⊞{k}_{n}}∕dx$ converges uniformly, as well as in all ${L}^{p}$-norms for $p>1$, to the density of $\nu$ except possibly in the neighborhood of one point. Applications include the global superconvergence to freely stable laws and that to free compound Poisson laws over the whole real line.

##### Keywords
freely infinitely divisible law, free convolution, superconvergence
Primary: 46L54
##### Milestones
Received: 9 October 2015
Revised: 1 July 2017
Accepted: 3 July 2017
Published: 17 October 2017
##### Authors
 Hari Bercovici Department of Mathematics Rawles Hall Indiana University Bloomington, Indiana 47405 United States Jiun-Chau Wang Department of Mathematics and Statistics University of Saskatchewan Saskatoon, S7N 5E6 Canada Ping Zhong Department of Pure Mathematics University of Waterloo Waterloo Ontario N2L 3G1 Canada School of Mathematics and Statistics Wuhan University Hubei China 430072