Fourth Moment Theorem and q-Brownian Chaos


Abstract: In 2005, Nualart and Peccati [13] 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. [9] extended this theorem to a sequence of normalized multiple Wigner integrals, in the context of the free Brownian motion. The q-Brownian motion, q ∈ (−1, 1], introduced by the physicists Frisch and Bourret [6] in 1970 and mathematically studied by Bo˙zejko and Speicher [2] 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

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