Invariance principles for homogeneous sums of free random variables

Abstract

We extend, in the free probability framework, an invariance principle for multilinear homogeneous sums with low influences recently established by Mossel, O’Donnel and Oleszkiewicz in [Ann. of Math. (2) 171 (2010) 295–341]. We then deduce several universality phenomenons, in the spirit of the paper [Ann. Probab. 38 (2010) 1947–1985] by Nourdin, Peccati and Reinert.