Central Limit for the Product of Free Random Variables

The central limit for the product of free random variables are studied by evaluating all the moments of the limit distribution. The logarithm of the central limit is found to be the same as the sum of two independent free random variables: one semicircularly distributed and another uniformly distributed. The logarithm of central limit has a moment-generating function of $\exp(ξ^2 s/2) {_{1}F_{1}}\left(1-s; 2; -ξ^2 s \right)$.

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