Open Access
Translator Disclaimer
2012 Convergence in law in the second Wiener/Wigner chaos
Ivan Nourdin, Guillaume Poly
Author Affiliations +
Electron. Commun. Probab. 17: 1-12 (2012). DOI: 10.1214/ECP.v17-2023


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.


Download Citation

Ivan Nourdin. Guillaume Poly. "Convergence in law in the second Wiener/Wigner chaos." Electron. Commun. Probab. 17 1 - 12, 2012.


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

zbMATH: 1261.60030
MathSciNet: MR2970700
Digital Object Identifier: 10.1214/ECP.v17-2023

Primary: 46L54
Secondary: 60F05 , 60G15 , 60H05

Keywords: convergence in law , Free probability , quadratic form , second Wiener chaos , second Wigner chaos


Back to Top