Limits Laws for Geometric Means of Free Positive Random Variables

Let {ak}k=1 be free identically distributed positive non–commuting random variables with probability measure distribution μ. In this paper we proved a multiplicative version of the Free Central Limit Theorem. More precisely, let bn = a 1/2 1 a 1/2 2 . . . an . . . a 1/2 2 a 1/2 1 then bn is a positive operator with the same moments as xn = a1a2 . . . an and b 1/2n n converges in distribution to positive operator Λ. We completely determined the probability measure distribution ν of Λ from the distribution μ. This gives us a natural map G : M+ → M+ with μ 7→ G(μ) = ν. 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 ν and the distribution of the Lyapunov exponents for the sequence {ak}k=1 introduced in [13].