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

We prove superconvergence results for all freely infinitely divisible distributions. Given a nondegenerate freely infinitely divisible distribution ν, let μn be a sequence of probability measures and let kn be a sequence of integers tending to infinity such that μnkn converges weakly to ν. We show that the density dμnkndx converges uniformly, as well as in all Lp-norms for p > 1, to the density of ν 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.

freely infinitely divisible law, free convolution, superconvergence
Mathematical Subject Classification 2010
Primary: 46L54
Received: 9 October 2015
Revised: 1 July 2017
Accepted: 3 July 2017
Published: 17 October 2017
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
Ping Zhong
Department of Pure Mathematics
University of Waterloo
Waterloo Ontario N2L 3G1
School of Mathematics and Statistics
Wuhan University
China 430072