Fourth Moment Theorem and q-Brownian Chaos
Abstract
In 2005, Nualart and Peccati showed the so-called Fourth Moment Theorem asserting that, for a sequence of normalized multiple Wiener-It\^o integrals to converge to the standard Gaussian law, it is necessary and sufficient that its fourth moment tends to 3. A few years later, Kemp et al. extended this theorem to a sequence of normalized multiple Wigner integrals, in the context of the free Brownian motion. The q-Brownian motion, q in (-1,1], introduced by the physicists Frisch and Bourret in 1970 and mathematically studied by Bozejko and Speicher in 1991, interpolates between the classical Brownian motion (q=1) and the free Brownian motion (q=0), and is one of the nicest examples of non-commutative processes. The question we shall solve in this paper is the following: what does the Fourth Moment Theorem become when dealing with a q-Brownian motion?
Keywords
Cite
@article{arxiv.1202.2545,
title = {Fourth Moment Theorem and q-Brownian Chaos},
author = {Aurélien Deya and Salim Noreddine and Ivan Nourdin},
journal= {arXiv preprint arXiv:1202.2545},
year = {2015}
}
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19 pages