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Strong asymptotic independence on Wiener chaos


Authors: Ivan Nourdin, David Nualart and Giovanni Peccati
Journal: Proc. Amer. Math. Soc. 144 (2016), 875-886
MSC (2010): Primary 60F05, 60H07, 60G15
DOI: https://doi.org/10.1090/proc12769
Published electronically: October 6, 2015
MathSciNet review: 3430861
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Abstract: Let $ F_n = (F_{1,n}, \dots ,F_{d,n})$, $ n\geq 1$, be a sequence of random vectors such that, for every $ j=1,\dots ,d$, the random variable $ F_{j,n}$ belongs to a fixed Wiener chaos of a Gaussian field. We show that, as $ n\to \infty $, the components of $ F_n$ are asymptotically independent if and only if $ \textup {Cov}(F_{i,n}^2,F_{j,n}^2)\to 0$ for every $ i\neq j$. Our findings are based on a novel inequality for vectors of multiple Wiener-Itô integrals, and represent a substantial refining of criteria for asymptotic independence in the sense of moments recently established by Nourdin and Rosiński (2014).


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Additional Information

Ivan Nourdin
Affiliation: Université du Luxembourg, Unité de Recherche en Mathématiques, 6 rue Richard Coudenhove-Kalergi, L-1359, Luxembourg
Email: ivan.nourdin@uni.lu

David Nualart
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045
Email: nualart@math.ku.edu

Giovanni Peccati
Affiliation: Université du Luxembourg, Unité de Recherche en Mathématiques, 6 rue Richard Coudenhove-Kalergi, L-1359, Luxembourg
Email: giovanni.peccati@gmail.com

DOI: https://doi.org/10.1090/proc12769
Keywords: Gaussian fields, independence, limit theorems, Malliavin calculus, Wiener chaos
Received by editor(s): January 8, 2014
Received by editor(s) in revised form: January 12, 2015
Published electronically: October 6, 2015
Additional Notes: The first author was partially supported by the ANR Grant ANR-10-BLAN-0121.
The second author was partially supported by the NSF grant DMS1208625.
The third author was partially supported by the grant F1R-MTH-PUL-12PAMP (PAMPAS), from Luxembourg University
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2015 American Mathematical Society