Poisson approximations on the free Wigner chaos. (English) Zbl 1281.46057
Summary: 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 I. Nourdin and G. Peccati [Ann. Probab. 37, No. 4, 1412–1426 (2009; Zbl 1171.60323)] and provides a noncentral counterpart to a result by T. Kemp et al. [ibid. 40, No. 4, 1577–1635 (2012; Zbl 1277.46033)]. 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.
MSC:
| 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 PDEs, etc.) |
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 chaosReferences:
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