Fourth Moment Theorem and q-Brownian Chaos

Abstract

In 2005, Nualart and Peccati (Ann Probab 33(1):177–193, 2005) proved the so-called Fourth Moment Theorem asserting that, for a sequence of normalized multiple Wiener-Itô 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. (Ann Probab 40(4):1577–1635, 2011) 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 (J Math Phys 11:364–390, 1970) in 1970 and mathematically studied by Bożejko and Speicher (Commun Math Phys 137:519–531, 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?