k-Divisible random variables in free probability

TL;DR

This paper introduces k-divisible elements in free probability, explores their properties, and develops tools for their analysis, including convolution formulas, infinite divisibility, and stable distributions, expanding the understanding of non-commutative random variables.

Contribution

It defines and studies k-divisible elements, deriving new formulas, convolution properties, and stability results in free probability theory.

Findings

Convolution with the zeta-function in NC relates to NC^k for k-divisible elements.

Free infinite divisibility is preserved under the k-th power mapping.

Reproducing properties for k-symmetric free stable distributions are established.

Abstract

We introduce and study the notion of k-divisible elements in a non-commutative probability space. A k-divisible element is a (non-commutative) random variable whose n-th moment vanishes whenever n is not a multiple of k. First, we consider the combinatorial convolution \ast in the lattices NC of non-crossing partitions and NC^k of k-divisible non-crossing partitions and show that convolving k times with the zeta-function in NC is equivalent to convolving once with the zeta-function in NC^k. Furthermore, when x is k-divisible, we derive a formula for the free cumulants of x^k in terms of the free cumulants of x, involving k-divisible non-crossing partitions. Second, we prove that if a and s are free and s is k-divisible then sps and a are free, whenever p is any polynomial (on a and s) of degree k - 2 on s. Moreover, we define a notion of R-diagonal k-tuples and prove similar results. Next, we show that free multiplicative convolution between a measure concentrated in the positive real line and a probability measure with k-symmetry is well defined. Analytic tools to calculate this convolution are developed. Finally, we concentrate on free additive powers of k-symmetric distributions and prove that \mu t is a well defined probability measure, for all t > 1. We derive central limit theorems and Poisson type ones. More generally, we consider freely infinitely divisible measures and prove that free infinite divisibility is maintained under the mapping \mu \rightarrow \mu^k . We conclude by focusing on (k-symmetric) free stable distributions, for which we prove a reproducing property generalizing the ones known for one sided and real symmetric free stable laws.