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Asymptotic behavior of the convex hull of a stationary Gaussian process*. / Davydov, Youri; Dombry, Clément.

In: Lithuanian Mathematical Journal, Vol. 52, No. 4, 10.2012, p. 363-368.

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Davydov, Y & Dombry, C 2012, 'Asymptotic behavior of the convex hull of a stationary Gaussian process*', Lithuanian Mathematical Journal, vol. 52, no. 4, pp. 363-368. https://doi.org/10.1007/s10986-012-9179-z

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Davydov, Youri ; Dombry, Clément. / Asymptotic behavior of the convex hull of a stationary Gaussian process*. In: Lithuanian Mathematical Journal. 2012 ; Vol. 52, No. 4. pp. 363-368.

BibTeX

@article{98ce694e5b5d4a7ca89767a61d80a5e2,
title = "Asymptotic behavior of the convex hull of a stationary Gaussian process*",
abstract = "Let X = {X(t), t ∈ T} be a stationary centered Gaussian process with values in ℝd, where the parameter set T equals ℕ or ℝ+. Let Σt = Cov(X0,Xt) be the covariance function of X, and (Ω,F,P) be the underlying probability space. We consider the asymptotic behavior of convex hulls Wt = conv{Xu, u ∈ T ∩ [0, t]} as t → +∞ and show that under the condition Σt → 0, t→∞, the rescaled convex hull (2 ln t)-1/2Wt converges almost surely (in the sense of Hausdorff distance) to an ellipsoid ε associated to the covariance matrix Σ0. The asymptotic behavior of the mathematical expectations Ef(Wt), where f is a homogeneous function, is also studied. These results complement and generalize in some sense the results of Davydov [Y. Davydov, On convex hull of Gaussian samples, Lith. Math. J., 51(2): 171-179, 2011].",
keywords = "convex hull, Gaussian processes, limit shape, limit theorem, stationary processes",
author = "Youri Davydov and Cl{\'e}ment Dombry",
note = "Funding Information: ∗ This research was supported in part by GDR grant 3477 “G{\'e}om{\'e}trie al{\'e}atoire.” Copyright: Copyright 2021 Elsevier B.V., All rights reserved.",
year = "2012",
month = oct,
doi = "10.1007/s10986-012-9179-z",
language = "English",
volume = "52",
pages = "363--368",
journal = "Lithuanian Mathematical Journal",
issn = "0363-1672",
publisher = "Springer Nature",
number = "4",

}

RIS

TY - JOUR

T1 - Asymptotic behavior of the convex hull of a stationary Gaussian process*

AU - Davydov, Youri

AU - Dombry, Clément

N1 - Funding Information: ∗ This research was supported in part by GDR grant 3477 “Géométrie aléatoire.” Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2012/10

Y1 - 2012/10

N2 - Let X = {X(t), t ∈ T} be a stationary centered Gaussian process with values in ℝd, where the parameter set T equals ℕ or ℝ+. Let Σt = Cov(X0,Xt) be the covariance function of X, and (Ω,F,P) be the underlying probability space. We consider the asymptotic behavior of convex hulls Wt = conv{Xu, u ∈ T ∩ [0, t]} as t → +∞ and show that under the condition Σt → 0, t→∞, the rescaled convex hull (2 ln t)-1/2Wt converges almost surely (in the sense of Hausdorff distance) to an ellipsoid ε associated to the covariance matrix Σ0. The asymptotic behavior of the mathematical expectations Ef(Wt), where f is a homogeneous function, is also studied. These results complement and generalize in some sense the results of Davydov [Y. Davydov, On convex hull of Gaussian samples, Lith. Math. J., 51(2): 171-179, 2011].

AB - Let X = {X(t), t ∈ T} be a stationary centered Gaussian process with values in ℝd, where the parameter set T equals ℕ or ℝ+. Let Σt = Cov(X0,Xt) be the covariance function of X, and (Ω,F,P) be the underlying probability space. We consider the asymptotic behavior of convex hulls Wt = conv{Xu, u ∈ T ∩ [0, t]} as t → +∞ and show that under the condition Σt → 0, t→∞, the rescaled convex hull (2 ln t)-1/2Wt converges almost surely (in the sense of Hausdorff distance) to an ellipsoid ε associated to the covariance matrix Σ0. The asymptotic behavior of the mathematical expectations Ef(Wt), where f is a homogeneous function, is also studied. These results complement and generalize in some sense the results of Davydov [Y. Davydov, On convex hull of Gaussian samples, Lith. Math. J., 51(2): 171-179, 2011].

KW - convex hull

KW - Gaussian processes

KW - limit shape

KW - limit theorem

KW - stationary processes

UR - http://www.scopus.com/inward/record.url?scp=84870904590&partnerID=8YFLogxK

U2 - 10.1007/s10986-012-9179-z

DO - 10.1007/s10986-012-9179-z

M3 - Article

AN - SCOPUS:84870904590

VL - 52

SP - 363

EP - 368

JO - Lithuanian Mathematical Journal

JF - Lithuanian Mathematical Journal

SN - 0363-1672

IS - 4

ER -

ID: 73460162