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Proceedings of the American Mathematical Society

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Almost sure convergence on chaoses

Authors: Guillaume Poly and Guangqu Zheng
Journal: Proc. Amer. Math. Soc. 147 (2019), 4055-4065
MSC (2010): Primary 60F99; Secondary 60H05
Published electronically: May 1, 2019
MathSciNet review: 3993797
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We present several new phenomena about almost sure convergence on homogeneous chaoses that include Gaussian Wiener chaos and homogeneous sums in independent random variables. Concretely, we establish the fact that almost sure convergence on a fixed finite sum of chaoses forces the almost sure convergence of each chaotic component. Our strategy uses ``extra randomness'' and a simple conditioning argument. These ideas are close to the spirit of Stein's method of exchangeable pairs. Some natural questions are left open in this note.

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Guillaume Poly
Affiliation: Institut de recherche mathématique de Rennes (UMR CNRS 6625), Université de Rennes 1, Bâtiment 22–23, 263, avenue du Général Leclerc, 35042 Rennes Cedex, France

Guangqu Zheng
Affiliation: Department of Mathematics, University of Kansas, Snow Hall, 1460 Jayhawk Boulevard, Lawrence, Kansas 66045-7594

Keywords: Exchangeable pairs, Ornstein-Uhlenbeck semigroup, Wiener chaos, homogeneous sums.
Received by editor(s): October 2, 2018
Received by editor(s) in revised form: December 13, 2018
Published electronically: May 1, 2019
Communicated by: Zhen-Qing Chen
Article copyright: © Copyright 2019 American Mathematical Society