2019
DOI: 10.1051/ps/2019013
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A Peccati-Tudor type theorem for Rademacher chaoses

Abstract: In this article, we prove that in the Rademacher setting, a random vector with chaotic components is close in distribution to a centred 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 (2017).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 est… Show more

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Cited by 9 publications
(6 citation statements)
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“…(5.4) can be shown in a similar manner to the proof of [59, Theorem 1.1] but using Lemmas 5.3 and 5.4 instead of Lemmas 2.1 and 3.1 in [59], respectively.…”
Section: Proof Of Proposition 52mentioning
confidence: 80%
See 1 more Smart Citation
“…(5.4) can be shown in a similar manner to the proof of [59, Theorem 1.1] but using Lemmas 5.3 and 5.4 instead of Lemmas 2.1 and 3.1 in [59], respectively.…”
Section: Proof Of Proposition 52mentioning
confidence: 80%
“…However, this leads to a bound containing the quantity Cov[F 2 , G 2 ] − 2 E [FG] 2 , so we need an additional argument to estimate it. For this reason, we take an alternative route for the proof, which is inspired by the discussions in Zheng [59] as well as [9,Proposition 3.6]. As a byproduct of this strategy, we obtain inequality (5.11) which leads to the universality of gamma variables.…”
Section: A Bound For the Variance Of The Carré Du Champ Operatormentioning
confidence: 99%
“…Now, let us present a simple path that leads to the fourth-moment-influence bound in Kolmogorov distance [9, Theorem 1.1] already mentioned in the introduction. For the bound in Wasserstein distance, we refer interested readers to [35] for a simple proof using exchangeable pairs. Suppose that F " J m pf q P L 4 pΩq for some f P H dm 0 and m P N such that ErF 2 s " 1, then it has been pointed out in the proof of [9,Lemma 3.7] that the random sequence u " pu k q kPN , defined as in (3.15), satisfies condition (2.14) in [17], which implies u P Dompδq.…”
Section: Normal Approximation Bounds For Rademacher Functionalsmentioning
confidence: 99%
“…Remark 2.5. Using the exchangeable pairs coupling constructed in Zheng (2019), it will also be possible to derive a multi-dimensional "fourth-moment-influence" type Wasserstein bound in the Rademacher setting via Theorem 1.3. We omit the details.…”
Section: Poisson Functionalsmentioning
confidence: 99%