Fourth Moment Theorem and q-Brownian Chaos

TL;DR

This paper extends the Fourth Moment Theorem to the setting of q-Brownian motion, a non-commutative process interpolating between classical and free Brownian motions, exploring convergence criteria in this context.

Contribution

It generalizes the Fourth Moment Theorem to q-Brownian motion, providing new insights into non-commutative probability and stochastic processes.

Findings

Established a version of the Fourth Moment Theorem for q-Brownian motion.

Identified conditions under which sequences of q-Wigner integrals converge to the q-Gaussian law.

Connected classical, free, and q-deformed probabilistic frameworks.

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?