CENTRAL LIMIT THEOREM FOR TENSOR PRODUCTS OF FREE VARIABLES

We establish a central limit theorem for tensor product random variables c_k := a_k ⊗ a_k, where (a_k)_k∈_N is a free family of variables. We show that if the variables a_k are centered, the limiting law is the semi-circle. Otherwise, the limiting law depends on the mean and variance of the variables a_k and corresponds to a free interpolation between the semi-circle law and the classical convolution of two semi-circle laws.