High-Dimensional Central Limit Theorems for Homogeneous Sums

Koike, Yuta

doi:10.1007/s10959-022-01156-2

Cited by 5 publications

(3 citation statements)

References 52 publications

(78 reference statements)

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“…The presence of the factor (log m) 1/4 is consistent with the fact that, for the standard Gaussian measure on R m , the isoperimetric constant associated with all hyper-rectangles of R m is bounded from above by √ log m, see [3,14]. An estimate analogous to (23) is established by different methods in [12,Corollary 3.1].…”

Section: A Proof and Discussion Of Relation (3)mentioning

confidence: 65%

Multivariate normal approximation on the Wiener space: new bounds in the convex distance

Nourdin

Peccati

Yang

2020

Preprint

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We establish explicit bounds on the convex distance between the distribution of a vector of smooth functionals of a Gaussian field, and that of a normal vector with a positive definite covariance matrix. Our bounds are commensurate to the ones obtained by Nourdin, Peccati and Réveillac (2010) for the (smoother) 1-Wasserstein distance, and do not involve any additional logarithmic factor. One of the main tools exploited in our work is a recursive estimate on the convex distance recently obtained by Schulte and Yukich (2019). We illustrate our abstract results in two different situations: (i) we prove a quantitative multivariate fourth moment theorem for vectors of multiple Wiener-Itô integrals, and (ii) we characterise the rate of convergence for the finite-dimensional distributions in the functional Breuer-Major theorem.

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Section: A Proof and Discussion Of Relation (3)mentioning

confidence: 65%

Multivariate normal approximation on the Wiener space: new bounds in the convex distance

Nourdin

Peccati

Yang

2020

Preprint

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“…Since we have max Thus we obtain (2.3). Finally, we have by Lemma 2.1 in Koike (2019) tr(A 4 j ) | W 4 j − 3( W 2 j ) 2 | + C 0 M A j 2 H.S. M(A j )…”

Section: Proof Of Applicationsmentioning

confidence: 84%

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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“…We can modify Assumption 2(i) to

for any

, a standard assumption in the literature on ultra-high-dimensional data analysis. This assumption ensures subexponential upper bounds for the tail probabilities of the statistics in question when

, as discussed in [ 23 , 24 ]. The requirement of sub-Gaussian properties in Assumption 2(i) is made for the sake of simplicity.…”

Section: Theoretical Resultsmentioning

confidence: 99%

A Blockwise Bootstrap-Based Two-Sample Test for High-Dimensional Time Series

Yang

2024

Entropy

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We propose a two-sample testing procedure for high-dimensional time series. To obtain the asymptotic distribution of our ℓ∞-type test statistic under the null hypothesis, we establish high-dimensional central limit theorems (HCLTs) for an α-mixing sequence. Specifically, we derive two HCLTs for the maximum of a sum of high-dimensional α-mixing random vectors under the assumptions of bounded finite moments and exponential tails, respectively. The proposed HCLT for α-mixing sequence under bounded finite moments assumption is novel, and in comparison with existing results, we improve the convergence rate of the HCLT under the exponential tails assumption. To compute the critical value, we employ the blockwise bootstrap method. Importantly, our approach does not require the independence of the two samples, making it applicable for detecting change points in high-dimensional time series. Numerical results emphasize the effectiveness and advantages of our method.

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High-Dimensional Central Limit Theorems for Homogeneous Sums

Cited by 5 publications

References 52 publications

Multivariate normal approximation on the Wiener space: new bounds in the convex distance

Multivariate normal approximation on the Wiener space: new bounds in the convex distance

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

A Blockwise Bootstrap-Based Two-Sample Test for High-Dimensional Time Series

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