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Convergence in law in the second Wiener/Wigner chaos


 
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1. Title Title of document Convergence in law in the second Wiener/Wigner chaos
 
2. Creator Author's name, affiliation, country Ivan Nourdin; Université de Lorraine; France
 
2. Creator Author's name, affiliation, country Guillaume Poly; Université Paris Est; France
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Convergence in law; second Wiener chaos; second Wigner chaos; quadratic form; free probability
 
3. Subject Subject classification 46L54; 60F05; 60G15; 60H05
 
4. Description 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Agence Nationale de la Recherche
 
7. Date (YYYY-MM-DD) 2012-08-18
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/2023
 
10. Identifier Digital Object Identifier 10.1214/ECP.v17-2023
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 17
 
12. Language English=en en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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