## The Annals of Probability

### 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 $\varphi(F_{n}^{4})-2\varphi(F_{n}^{3})\rightarrow2\lambda^{2}-\lambda$ (where $\varphi$ 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.

#### Article information

Source
Ann. Probab., Volume 41, Number 4 (2013), 2709-2723.

Dates
First available in Project Euclid: 3 July 2013

https://projecteuclid.org/euclid.aop/1372859764

Digital Object Identifier
doi:10.1214/12-AOP815

Mathematical Reviews number (MathSciNet)
MR3112929

Zentralblatt MATH identifier
1281.46057

#### Citation

Nourdin, Ivan; Peccati, Giovanni. Poisson approximations on the free Wigner chaos. Ann. Probab. 41 (2013), no. 4, 2709--2723. doi:10.1214/12-AOP815. https://projecteuclid.org/euclid.aop/1372859764

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