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Zapiski Nauchnykh Seminarov POMI, 2022, Volume 510, Pages 87–97
(Mi znsl7195)
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More on the convergence of Gaussian convex hulls
Yu. Davydovab, V. Paulauskasc a Université de Lille, Laboratoire Paul Painlevé, 42 rue Paul Duez 59000 Lille - France
b Saint Petersburg state university, 7-9 Universitetskaya Embankment, St Petersburg, Russia
c Vilnius University, Department of Mathematics and Informatics, Naugarduko st. 24, LT-03225, Vilnius, Lithuania
Abstract:
A “law of large numbers” for consecutive convex hulls for weakly dependent Gaussian sequences $\{X_n\}$, having the same marginal distribution, is extended to the case when the sequence $\{X_n\}$ has a weak limit. Let $\mathbb{B}$ be a separable Banach space with a conjugate space $\mathbb{B}^\ast$. Let $\{X_n\}$ be a centered $\mathbb{B}$-valued Gaussian sequence satisfying two conditions: 1) $X_n \Rightarrow X $ and 2) For every $x^* \in \mathbb{B}^\ast$ $$ \lim_ {n,m, |n-m|\rightarrow \infty}E\langle X_n, x^*\rangle \langle X_m, x^*\rangle = 0. $$ Then with probability $1$ the normalized convex hulls $$ W_n = \frac{1}{(2\ln n)^{1/2}} \mathrm{conv} \{ X_1,\ldots,X_{n} \} $$ converge in Hausdorff distance to the concentration ellipsoid of a limit Gaussian $\mathbb{B}$-valued random element $X.$ In addition, some related questions are discussed.
Key words and phrases:
Gaussian sequences, convex hull, limit behavior.
Received: 25.07.2022
Citation:
Yu. Davydov, V. Paulauskas, “More on the convergence of Gaussian convex hulls”, Probability and statistics. Part 32, Zap. Nauchn. Sem. POMI, 510, POMI, St. Petersburg, 2022, 87–97
Linking options:
https://www.mathnet.ru/eng/znsl7195 https://www.mathnet.ru/eng/znsl/v510/p87
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Abstract page: | 37 | Full-text PDF : | 12 | References: | 13 |
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