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].

Published in Bernoulli (http://dx.doi.org/10.3150/12-BEJ498) by the International Statistical Institute/Bernoulli Society

References in corpus (4)

• Central limit theorems for sequences of multiple stochastic integrals
• Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos
• Wigner chaos and the fourth moment
• Convergence of Wigner integrals to the tetilla law

Cited by in corpus (8)

• Poisson convergence on the free Poisson algebra
• Lectures on Gaussian approximations with Malliavin calculus
• Homoscedasticity tests for both low and high-dimensional fixed design regressions
• Universality of free homogeneous sums in every dimension
• Berry-Esseen bounds for the multivariate $\mathcal{B}$-free CLT and operator-valued matrices
• Multidimensional limit theorems for homogeneous sums: a general transfer principle
• Operator-Valued Matrices with Free or Exchangeable Entries
• A Moment Majorization principle for random matrix ensembles