Central Limit for the Product of Free Random Variables

TL;DR

This paper investigates the central limit theorem for the product of free random variables, revealing that its logarithm equals the sum of a semicircular and a uniform free variable, with a specific moment-generating function.

Contribution

It introduces a novel analysis of the central limit for free variable products, characterizing the distribution of the logarithm as a sum of two specific free distributions.

Findings

Logarithm of the limit distribution matches the sum of a semicircular and a uniform free variable.

Derived the moment-generating function for the logarithm of the limit distribution.

Evaluated all moments of the limit distribution for the product of free random variables.

Abstract

The central limit for the product of free random variables are studied by evaluating all the moments of the limit distribution. The logarithm of the central limit is found to be the same as the sum of two independent free random variables: one semicircularly distributed and another uniformly distributed. The logarithm of central limit has a moment-generating function of $\exp(\xi^2 s/2) {_{1}F_{1}}\left(1-s; 2; -\xi^2 s \right)$.