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.