Open Access
Translator Disclaimer
July 2013 Poisson approximations on the free Wigner chaos
Ivan Nourdin, Giovanni Peccati
Ann. Probab. 41(4): 2709-2723 (July 2013). DOI: 10.1214/12-AOP815


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.


Download Citation

Ivan Nourdin. Giovanni Peccati. "Poisson approximations on the free Wigner chaos." Ann. Probab. 41 (4) 2709 - 2723, July 2013.


Published: July 2013
First available in Project Euclid: 3 July 2013

zbMATH: 1281.46057
MathSciNet: MR3112929
Digital Object Identifier: 10.1214/12-AOP815

Primary: 46L54 , 60H05 , 60H07 , 60H30

Keywords: Catalan numbers , contractions , free Brownian motion , free cumulants , free Poisson distribution , Free probability , Marchenko–Pastur , noncentral limit theorems , noncrossing partitions , Riordan numbers , semicircular distribution , Wigner chaos

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.41 • No. 4 • July 2013
Back to Top