On the rate of convergence for central limit theorems of sojourn times of Gaussian fields

Pham, Viet-Hung

doi:10.1016/j.spa.2013.01.016

“…In the paper Pham (2013), the author has studied the rate of convergence for the central limit theorems in the case of the fixed level and moving level using Wasserstein distance d W . The central limit theorems are based on the following assumption:…”

Section: Sojourn Times Of Gaussian Fieldsmentioning

confidence: 99%

“…In the paper Pham (2013), the author takes N T such that 3 N T = T −β+ d 2 . This N T , by using Theorem 6, yields the explicit upper bound of the Kolmogorov distance:…”

Section: Remark 3 (A) If We Choose N T Such Thatmentioning

confidence: 99%

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Kolmogorov distance for the central limit theorems of the Wiener chaos expansion and applications

Kim

¹

,

Park

²

2015

Journal of the Korean Statistical Society

2

1

0

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a b s t r a c tThis paper concerns the rate of convergence for the central limit theorems of the chaos expansion of functionals of Gaussian process. The aim of the present work is to derive upper bounds of the Kolmogorov distance for the rate of convergence. We apply our results to find the upper bound of the Kolmogorov distance in the quantitative Breuer-Major theorems (Nourdin et al., 2011), and prove that the upper bound in our results is more efficient than that in the quantitative Breuer-Major theorems. Also we obtain the explicit upper bound of the Kolmogorov distance for central limit theorems of sojourn times.

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“…Let us now reconsider the previous examples. [17,19,24]) Using the notation introduced in Example 9, we get that…”

Section: A Simple Example Of Non-gaussian Eigenfunctions Is Provided Bymentioning

confidence: 99%

Approximate normality of high-energy hyperspherical eigenfunctions

Campese

¹

,

Marinucci

²

,

Rossi

³

2018

Journal of Mathematical Analysis and Applications

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The Berry heuristic has been a long standing ansatz about the high energy (i.e. large eigenvalues) behaviour of eigenfunctions (see [8]). Roughly speaking, it states that under some generic boundary conditions, these eigenfunctions exhibit Gaussian behaviour when the eigenvalues grow to infinity. Our aim in this paper is to make this statement quantitative and to establish some rigorous bounds on the distance to Gaussianity, focussing on the hyperspherical case (i.e., for eigenfunctions of the Laplace-Beltrami operator on the normalized d-dimensional sphere -also known as spherical harmonics). Some applications to non-Gaussian models are also discussed.•

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“…

Our interest in this paper is to explore limit theorems for various geometric functionals of excursion sets of isotropic Gaussian random fields. In the past, limit theorems have been proven for various geometric functionals of excursion sets/sojourn times ( see [5,15,16,19,24,28] for a sample of works in such settings). The most recent addition being [8] where a CLT for Euler-Poincaré characteristic of the excursions set of a Gaussian random field is proven under appropriate conditions.In this paper, we shall obtain a central limit theorem for some global geometric functionals, called the Lipschitz-Killing curvatures of excursion sets of Gaussian random fields in an appropriate setting.the excursion set of f over a threshold u, denoted by A u (f ; T ), asOur interest, in this paper, is to study the distributional aspects of Lipschitz-Killing curvatures of the sets A u (f ; T ).The Lipschitz-Killing curvatures (LKCs) of a d-dimensional Whitney stratified manifold 1 M are (d + 1) integral geometric functionals {L k (M )} d k=0 , with L 0 (M ) the Euler-Poincaré characteristic of the set M , and L d (M ) the d-dimensional Hausdorff measure of M .

…”

mentioning

confidence: 99%

CLT for Lipschitz-Killing Curvatures of Excursion Sets of Gaussian Random Fields

Kratz

¹

,

Vadlamani

²

2016

SSRN Journal

2

0

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Our interest in this paper is to explore limit theorems for various geometric functionals of excursion sets of isotropic Gaussian random fields. In the past, limit theorems have been proven for various geometric functionals of excursion sets/sojourn times ( see [5,15,16,19,24,28] for a sample of works in such settings). The most recent addition being [8] where a CLT for Euler-Poincaré characteristic of the excursions set of a Gaussian random field is proven under appropriate conditions.In this paper, we shall obtain a central limit theorem for some global geometric functionals, called the Lipschitz-Killing curvatures of excursion sets of Gaussian random fields in an appropriate setting.the excursion set of f over a threshold u, denoted by A u (f ; T ), asOur interest, in this paper, is to study the distributional aspects of Lipschitz-Killing curvatures of the sets A u (f ; T ).The Lipschitz-Killing curvatures (LKCs) of a d-dimensional Whitney stratified manifold 1 M are (d + 1) integral geometric functionals {L k (M )} d k=0 , with L 0 (M ) the Euler-Poincaré characteristic of the set M , and L d (M ) the d-dimensional Hausdorff measure of M . Though, for k = 1, . . . , d − 1, the L k (M ) do not have such clear interpretation, the scaling property 2 of the LKCs can be used to interpret the k-th LKC as a k-dimensional measure. This property not only underlines the importance of the LKCs, but also characterises them with their additive and scaling property together with the rigid motion invariance.

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On the rate of convergence for central limit theorems of sojourn times of Gaussian fields

Cited by 20 publications

References 13 publications

Kolmogorov distance for the central limit theorems of the Wiener chaos expansion and applications

Kolmogorov distance for the central limit theorems of the Wiener chaos expansion and applications

Approximate normality of high-energy hyperspherical eigenfunctions

CLT for Lipschitz-Killing Curvatures of Excursion Sets of Gaussian Random Fields

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