The Annals of Probability

Poisson approximations on the free Wigner chaos

Ivan Nourdin and Giovanni Peccati

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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.

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

First available in Project Euclid: 3 July 2013

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Zentralblatt MATH identifier

Primary: 46L54: Free probability and free operator algebras 60H05: Stochastic integrals 60H07: Stochastic calculus of variations and the Malliavin calculus 60H30: Applications of stochastic analysis (to PDE, etc.)

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


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

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