Poisson approximations on the free Wigner chaos
Résumé
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 free Poisson random variable with rate lambda>0 if and only if varphi(F_n^4)- 2 varphi(F_n^3)--> 2 lambda^2-lambda (where varphi is the relevant tracial operator). This extends to a free setting some recent limit theorems by Nourdin and Peccati (2009), and provides a non-central counterpart to a result by Kemp et al. (2011). As a by-product of our findings, we show that Wigner chaoses of order strictly greater than 2 do not contain non-zero free Poisson random variables. Our techniques involve the so-called 'Riordan numbers', counting non-crossing partitions without singletons.
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