## Electronic Communications in Probability

### Convergence in law in the second Wiener/Wigner chaos

#### Abstract

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 $F_\infty$ 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 $F_\infty$. 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.

An Erratum is available in ECP volume 17 paper number 54.

#### Article information

Source
Electron. Commun. Probab., Volume 17 (2012), paper no. 36, 12 pp.

Dates
Accepted: 18 August 2012
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465263169

Digital Object Identifier
doi:10.1214/ECP.v17-2023

Mathematical Reviews number (MathSciNet)
MR2970700

Zentralblatt MATH identifier
1261.60030

Rights

#### Citation

Nourdin, Ivan; Poly, Guillaume. Convergence in law in the second Wiener/Wigner chaos. Electron. Commun. Probab. 17 (2012), paper no. 36, 12 pp. doi:10.1214/ECP.v17-2023. https://projecteuclid.org/euclid.ecp/1465263169

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