Poisson convergence on the free Poisson algebra
Abstract
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.
Keywords
Cite
@article{arxiv.1312.2610,
title = {Poisson convergence on the free Poisson algebra},
author = {Solesne Bourguin},
journal= {arXiv preprint arXiv:1312.2610},
year = {2015}
}
Comments
Published at http://dx.doi.org/10.3150/14-BEJ638 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)