IUMJ

Title: Limits laws for geometric means of free random variables

Authors: Gabriel H. Tucci

Issue: Volume 59 (2010), Issue 1, 1-14

Abstract:

Let $\{T_{k}\}_{k=1}^{\infty}$ be a family of $*$-free identically distributed operators in a finite von Neumann algebra. In this work we prove a multiplicative version of the Free Central Limit Theorem. More precisely, let $B_{n} = T_{1}^{*} T_{2}^{*} \dots T_{n}^{*} T_{n} \dots T_{2}T_{1}$; then $B_{n}$ is a positive operator and $B_{n}^{1/2n}$ converges in distribution to an operator $\Lambda$. We completely determine the probability distribution $\nu$ of $\Lambda$ from the distribution $\mu$ of $|T|^{2}$. This gives us a natural map $\mathcal{G}: \mathcal{M}_{+} \to \mathcal{M}_{+}$ with $\mu \mapsto \mathcal{G}(\mu) = \nu$. We study how this map behaves with respect to additive and multiplicative free convolution. As an interesting consequence of our results, we illustrate the relation between the probability distribution $\nu$ and the distribution of the Lyapunov exponents for the sequence $\{T_{k}\}_{k=1}^{\infty}$ introduced in [V. Kargin, \textit{Lyapunov exponents of free operators}, J. Funct. Anal. \textbf{255} (2008), 1874--1888).