Local limit theorems in free probability theory

TL;DR

This paper investigates the superconvergence phenomena in free probability, demonstrating uniform and $L^p$-convergence of densities to the semicircular law and deriving an entropic central limit theorem.

Contribution

It establishes new convergence results for densities in the free central limit theorem, including uniform and $L^p$-convergence, and introduces an entropic CLT.

Findings

Densities converge uniformly to the semicircular density.

Densities also converge in $L^p$ norm for $p>1/2$.

An entropic central limit theorem is proved.

Abstract

In this paper, we study the superconvergence phenomenon in the free central limit theorem for identically distributed, unbounded summands. We prove not only the uniform convergence of the densities to the semicircular density but also their $L^p$-convergence to the same limit for $p>1/2$. Moreover, an entropic central limit theorem is obtained as a consequence of the above results.