Poisson approximations on the free Wigner chaos

Abstract

We prove that an adequately rescaled sequence $\{F_n\}$ of self-adjoint operators, living inside a fixed free Wigner chaos of even order, converges in distribution to a centered free Poisson random variable with rate $\lambda >0$ if and only if $\phi (F_n^4)-2\phi (F_n^3)\to 2{\lambda }^2-\lambda$ (where $\phi$ is the relevant tracial state). This extends to a free setting some recent limit theorems by Nourdin and Peccati [Ann. Probab. 37 (2009) 1412-1426] and provides a noncentral counterpart to a result by Kemp et al. [Ann. Probab. 40 (2012) 1577-1635]. As a by-product of our findings, we show that Wigner chaoses of order strictly greater than 2 do not contain nonzero free Poisson random variables. Our techniques involve the so-called "Riordan numbers," counting noncrossing partitions without singletons.

Published in at http://dx.doi.org/10.1214/12-AOP815 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Central limit theorems for sequences of multiple stochastic integrals
• Noncentral convergence of multiple integrals
• Wigner chaos and the fourth moment
• Fourth Moment Theorem and q-Brownian Chaos
• Convergence of Wigner integrals to the tetilla law

Cited by in corpus (3)

• Convergence in law in the second Wiener/Wigner chaos
• Poisson convergence on the free Poisson algebra
• Freeness characterizations on free chaos spaces