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 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].
Keywords
Cite
@article{arxiv.1201.1753,
title = {Invariance principles for homogeneous sums of free random variables},
author = {Aurélien Deya and Ivan Nourdin},
journal= {arXiv preprint arXiv:1201.1753},
year = {2014}
}
Comments
Published in Bernoulli (http://dx.doi.org/10.3150/12-BEJ498) by the International Statistical Institute/Bernoulli Society