Research output: Contribution to journal › Article › peer-review
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.Research output: Contribution to journal › Article › peer-review
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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