Poisson approximations on the free Wigner chaos

Autores: Ivan Nourdin, Giovanni Peccati
Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 4, 2013, págs. 2709-2723
Idioma: inglés
DOI: 10.1214/12-AOP815
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Resumen
- We prove that an adequately rescaled sequence {Fn} 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 λ>0 if and only if φ(F4n)−2φ(F3n)→2λ2−λ (where φ 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.