Limits Laws for Geometric Means of Free Random Variables

Let {Tk}(k=1)(infinity) 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)*...T(n)*Tn...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 p of ITV. This gives us a natural map G : M(+) -> M(+) with mu -> 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 v and the distribution of the Lyapunov exponents for the sequence {T(k)}(k=1)(infinity) introduced in [12].