On Asymptotic Growth of the Support of Free Multiplicative Convolutions

Abstract

Let $\mu$ be a compactly supported probability measure on $\mathbb{R}^{+}$ with expectation $1$ and variance $V.$ Let $\mu _{n}$ denote the $n$-time free multiplicative convolution of measure $\mu $ with itself. Then, for large $n$ the length of the support of $\mu _{n}$ is asymptotically equivalent to $eVn$, where $e$ is the base of natural logarithms, $ e=2.71\ldots $.