- Volume 21, Number 4 (2015), 2139-2156.
Poisson convergence on the free Poisson algebra
Based on recent findings by Bourguin and Peccati, we give a fourth moment type condition for an element of a free Poisson chaos of arbitrary order to converge to a free (centered) Poisson distribution. We also show that free Poisson chaos of order strictly greater than one do not contain any non-zero free Poisson random variables. We are also able to give a sufficient and necessary condition for an element of the first free Poisson chaos to have a free Poisson distribution. Finally, depending on the parity of the considered free Poisson chaos, we provide a general counterexample to the naive universality of the semicircular Wigner chaos established by Deya and Nourdin as well as a transfer principle between the Wigner and the free Poisson chaos.
Bernoulli Volume 21, Number 4 (2015), 2139-2156.
Received: December 2013
Revised: April 2014
First available in Project Euclid: 5 August 2015
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Bourguin, Solesne. Poisson convergence on the free Poisson algebra. Bernoulli 21 (2015), no. 4, 2139--2156. doi:10.3150/14-BEJ638. https://projecteuclid.org/euclid.bj/1438777589.