A Peccati-Tudor type theorem for Rademacher chaoses

Zheng, Guangqu

doi:10.1051/ps/2019013

Cited by 9 publications

(6 citation statements)

References 23 publications

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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%

“…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%

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High-Dimensional Central Limit Theorems for Homogeneous Sums

Koike

2022

J Theor Probab

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This paper develops a quantitative version of de Jong’s central limit theorem for homogeneous sums in a high-dimensional setting. More precisely, under appropriate moment assumptions, we establish an upper bound for the Kolmogorov distance between a multi-dimensional vector of homogeneous sums and a Gaussian vector so that the bound depends polynomially on the logarithm of the dimension and is governed by the fourth cumulants and the maximal influences of the components. As a corollary, we obtain high-dimensional versions of fourth-moment theorems, universality results and Peccati–Tudor-type theorems for homogeneous sums. We also sharpen some existing (quantitative) central limit theorems by applications of our result.

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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%

Section: A Bound For the Variance Of The Carré Du Champ Operatormentioning

confidence: 99%

High-Dimensional Central Limit Theorems for Homogeneous Sums

Koike

2022

J Theor Probab

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“…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%

A simplified second-order Gaussian Poincaré inequality in discrete setting with applications

Eichelsbacher¹,

Rednoß²,

Thäle³

et al. 2021

Preprint

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In this paper, a simplified second-order Gaussian Poincaré inequality for normal approximation of functionals over infinitely many Rademacher random variables is derived. It is based on a new bound for the Kolmogorov distance between a general Rademacher functional and a Gaussian random variable, which is established by means of the discrete Malliavin-Stein method and is of independent interest. As an application, the number of vertices with prescribed degree and the subgraph counting statistic in the Erdős-Rényi random graph are discussed. The number of vertices of fixed degree is also studied for percolation on the Hamming hypercube. Moreover, the number of isolated faces in the Linial-Meshulam-Wallach random κ-complex and infinite weighted 2-runs are treated.

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“…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%

New error bounds in multivariate normal approximations via exchangeable pairs with applications to Wishart matrices and fourth moment theorems

Xiao¹,

Koike²

2020

Preprint

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We extend Stein's celebrated Wasserstein bound for normal approximation via exchangeable pairs to the multi-dimensional setting. As an intermediate step, we exploit the symmetry of exchangeable pairs to obtain an error bound for smooth test functions. We also obtain a continuous version of the multi-dimensional Wasserstein bound in terms of fourth moments. We apply the main results to multivariate normal approximations to Wishart matrices of size n and degree d, where we obtain the optimal convergence rate n 3 /d for smooth test functions under only moment assumptions, and to quadratic forms and Poisson functionals, where we strengthen a few of the fourth moment bounds in the literature on the Wasserstein distance.

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A Peccati-Tudor type theorem for Rademacher chaoses

Cited by 9 publications

References 23 publications

High-Dimensional Central Limit Theorems for Homogeneous Sums

High-Dimensional Central Limit Theorems for Homogeneous Sums

A simplified second-order Gaussian Poincaré inequality in discrete setting with applications

New error bounds in multivariate normal approximations via exchangeable pairs with applications to Wishart matrices and fourth moment theorems

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