Superconvergence and regularity of densities in free probability

TL;DR

This paper investigates superconvergence phenomena and regularity properties of densities in free probability, extending known results to broader classes of free convolutions and infinitely divisible laws.

Contribution

It demonstrates superconvergence for products of free, identically distributed variables and extends Holder regularity results to more general free convolutions.

Findings

Superconvergence occurs for products of free, identically distributed variables.

Holder regularity extends to free additive and multiplicative convolutions with arbitrary freely infinitely divisible laws.

Regularity results apply to free convolution semigroups.

Abstract

The superconvergence phenomenon is shown for products of free, identically distributed random variables. We also show that a certain Holder regularity, first demonstrated by Biane for the density of a free additive convolution with a semicircular law, extends to free additive and multiplicative convolutions with arbitrary freely infinitely divisible laws and to free convolution semigroups.