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Fourth Moment Theorem and q-Brownian Chaos

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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?

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Authors and Affiliations

  1. Institut Élie Cartan, Université de Lorraine, BP 70239, 54506, Vandoeuvre-lès-Nancy, France

    Aurélien Deya & Ivan Nourdin

  2. Laboratoire de Probabilités et Modèles Aléatoires, Université Paris 6, Boîte courrier 188, 4 Place Jussieu, 75252, Paris Cedex 5, France

    Salim Noreddine

Authors
  1. Aurélien Deya
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  2. Salim Noreddine
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  3. Ivan Nourdin
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Correspondence to Aurélien Deya.

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Communicated by Y. Kawahigashi

Supported in part by the two following (french) ANR grants: ‘Exploration des Chemins Rugueux’ [ANR-09-BLAN-0114] and ‘Malliavin, Stein and Stochastic Equations with Irregular Coefficients’ [ANR-10-BLAN-0121].

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Deya, A., Noreddine, S. & Nourdin, I. Fourth Moment Theorem and q-Brownian Chaos. Commun. Math. Phys. 321, 113–134 (2013). https://doi.org/10.1007/s00220-012-1631-8

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  • DOI: https://doi.org/10.1007/s00220-012-1631-8

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