Convergence in law in the second Wiener/Wigner chaos

Ivan Nourdin (1) , Guillaume Poly (2)

1 IECN - Institut Élie Cartan de Nancy
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

Let L be the class of limiting laws associated with sequences in the second Wiener chaos. We exhibit a large subset L_0 of L satisfying that, for any F_infinity in L_0, the convergence of only a finite number of cumulants suffices to imply the convergence in law of any sequence in the second Wiener chaos to F_infinity. This result is in the spirit of the seminal paper by Nualart and Peccati, in which the authors discovered the surprising fact that convergence in law for sequences of multiple Wiener-Itô integrals to the Gaussian is equivalent to convergence of just the fourth cumulant. Also, we offer analogues of this result in the case of free Brownian motion and double Wigner integrals, in the context of free probability.

Keywords: free probability.; free probability; Convergence in law; second Wiener chaos; second Wigner chaos; quadratic form

Domains: Probability [math.PR]

Ivan Nourdin, Guillaume Poly. Convergence in law in the second Wiener/Wigner chaos. Electronic Communications in Probability, 2012, 17 (36), pp.1-12. ⟨10.1214/ECP.v17-2023⟩. ⟨hal-00696501v3⟩