Superconvergence to freely infinitely divisible distributions

TL;DR

This paper proves that sums of free, infinitesimal, identically distributed variables rapidly develop smooth densities, converging uniformly and in various norms to a freely infinitely divisible law, with applications to stable and compound Poisson laws.

Contribution

It establishes the superconvergence phenomenon for all freely infinitely divisible distributions, showing rapid density regularization in free probability.

Findings

Sum distributions become absolutely continuous with smooth densities.

Densities approximate the limit law uniformly and in all L^p norms.

Results apply to free stable and free compound Poisson laws.

Abstract

The phenomenon of superconvergence is proved for all freely infinitely divisible distributions. Precisely, suppose that the partial sums of a sequence of free identically distributed, infinitesimal random variables converge in distribution to a nondegenerate freely infinitely divisible law. Then the distribution of the sum becomes Lebesgue absolutely continuous with a continuous density in finite time, and this density can be approximated by that of the limit law uniformly, as well as in all $L^{p}$-norms for $p>1$, on the real line 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.