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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Strong asymptotic independence on Wiener chaos
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by Ivan Nourdin, David Nualart and Giovanni Peccati PDF
Proc. Amer. Math. Soc. 144 (2016), 875-886 Request permission

Abstract:

Let $F_n = (F_{1,n}, \dots ,F_{d,n})$, $n\geqslant 1$, 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 $\mathrm {Cov}(F_{i,n}^2,F_{j,n}^2)\to 0$ 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).
References
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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
  • MR Author ID: 730973
  • Email: ivan.nourdin@uni.lu
  • David Nualart
  • Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045
  • MR Author ID: 132560
  • Email: nualart@math.ku.edu
  • Giovanni Peccati
  • Affiliation: Université du Luxembourg, Unité de Recherche en Mathématiques, 6 rue Richard Coudenhove-Kalergi, L-1359, Luxembourg
  • MR Author ID: 683104
  • Email: giovanni.peccati@gmail.com
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 875-886
  • MSC (2010): Primary 60F05, 60H07, 60G15
  • DOI: https://doi.org/10.1090/proc12769
  • MathSciNet review: 3430861