Open Access
November 2015 Poisson convergence on the free Poisson algebra
Solesne Bourguin
Bernoulli 21(4): 2139-2156 (November 2015). DOI: 10.3150/14-BEJ638


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.


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Solesne Bourguin. "Poisson convergence on the free Poisson algebra." Bernoulli 21 (4) 2139 - 2156, November 2015.


Received: 1 December 2013; Revised: 1 April 2014; Published: November 2015
First available in Project Euclid: 5 August 2015

zbMATH: 1375.46047
MathSciNet: MR3378462
Digital Object Identifier: 10.3150/14-BEJ638

Keywords: chaos structure , combinatorics of free Poisson random measures , contractions , diagram formulae , Fourth moment theorem , free Poisson distribution , Free probability , multiplication formula

Rights: Copyright © 2015 Bernoulli Society for Mathematical Statistics and Probability

Vol.21 • No. 4 • November 2015
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