Fourth moment theorem and q-Brownian chaos

Aurélien Deya (1) , Salim Noreddine (2) , Ivan Nourdin (1)

1 IECL - Institut Élie Cartan de Lorraine
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

In 2005, Nualart and Peccati showed 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. 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?

Mots clés: Central limit theorems; multiple integrals; q-Brownian motion; multiple integrals.; non-commutative probability space

Domaines: Probabilités [math.PR]; Algèbres d'opérateurs [math.OA]

Aurélien Deya, Salim Noreddine, Ivan Nourdin. Fourth moment theorem and q-Brownian chaos. Communications in Mathematical Physics, 2013, 321 (1), pp.113-134. ⟨10.1007/s00220-012-1631-8⟩. ⟨hal-00669247⟩