Superconvergence to freely infinitely divisible distributions

Hari Bercovici, Jiun-Chau Wang and Ping Zhong

Vol. 292 (2018), No. 2, 273–291

DOI: 10.2140/pjm.2018.292.273

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

Mathematical Subject Classification 2010

Primary: 46L54

Milestones

Received: 9 October 2015

Revised: 1 July 2017

Accepted: 3 July 2017

Published: 17 October 2017