POISSON APPROXIMATIONS ON THE FREE WIGNER CHAOS

• 成果类型: Article
• 署名作者: Nourdin, Ivan; Peccati, Giovanni
• 署名单位: Universite de Lorraine
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP815
• 发表日期: 2013
• 页码: 2709-2723
• 关键词: convergence

摘要

We prove that an adequately resealed 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 phi(F-n(4)) - 2 phi(F-n(3)) -> 2 lambda(2) - lambda (where phi 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.