Invariance principles for homogeneous sums of free random variables

TL;DR

This paper extends an invariance principle for multilinear homogeneous sums within free probability, establishing hypercontractivity and demonstrating universality phenomena, thus broadening the understanding of invariance in non-commutative probability spaces.

Contribution

It introduces a hypercontractivity property for homogeneous sums in free probability, enabling the extension of invariance principles and revealing universality phenomena.

Findings

Established hypercontractivity for homogeneous sums in free probability.

Extended invariance principles to free random variables.

Demonstrated universality phenomena in free probability context.

Abstract

We extend, in the free probability framework, an invariance principle for multilinear homogeneous sums with low influences recently established in [E. Mossel, R. O'Donnell and K. Oleszkiewicz (2010). Noise stability of functions with low influences: invariance and optimality. {\it Ann. Math.} {\bf 171}, no. 1, 295-341]. To do so, a hypercontractivity property for those homogeneous sums is necessary, and to prove it has turned out to be our main task. Finally, we deduce from our extension several universality phenomenons, in the spirit of the paper [I. Nourdin, G. Peccati and G. Reinert (2010). Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos. {\it Ann. Probab.} {\bf 38}, no. 5, 1947-1985].