Volume 23, 2019
|Page(s)||874 - 892|
|Published online||24 December 2019|
University of Kansas, Mathematics Department,
1460 Jayhawk Blvd,
Accepted: 24 June 2019
In this article, we prove that in the Rademacher setting, a random vector with chaotic components is close in distribution to a centered Gaussian vector, if both the maximal influence of the associated kernel and the fourth cumulant of each component is small. In particular, we recover the univariate case recently established in Döbler and Krokowski (2019). Our main strategy consists in a novel adaption of the exchangeable pairs couplings initiated in Nourdin and Zheng (2017), as well as its combination with estimates via chaos decomposition.
Mathematics Subject Classification: 60F05 / 60B12 / 47N30
Key words: Fourth moment theorem / Rademacher chaos / Stein’s method / exchangeable pairs / spectral decomposition / maximal influence
Part of this work was done during a visit at National University of Singapore. I thank very much Professor Louis H.Y. Chen at NUS for his very generous support and kind hospitality.
The gratitude also goes to Professor Giovanni Peccati for sharing his alternative proof of Lemma 2.4 in , which motived our proof of Lemma 2.1.
© EDP Sciences, SMAI 2019
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