Geodesic
More on the convergence of Gaussian convex hulls
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 32, Tome 510 (2022), pp. 87-97 
Yu. Davydov; V. Paulauskas. More on the convergence of Gaussian convex hulls. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 32, Tome 510 (2022), pp. 87-97. http://geodesic.mathdoc.fr/item/ZNSL_2022_510_a4/
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     title = {More on the convergence of {Gaussian} convex hulls},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.

[1] Yu. Davydov, “On convex hull of Gaussian samples”, Lith. Math. J., 51 (2011), 171–179 | DOI | MR

[2] Yu. Davydov, C. Dombry, “Asymptotic behavior of the convex hull of a stationary Gaussian process”, Lith. Math. J., 52:3 (2012), 363–368 | DOI | MR

[3] Yu. Davydov, V. Paulauskas, “On the asymptotic form of convex hulls of Gaussian random fields”, Cent. Eur. J. Math., 12:5 (2014), 711–720 | MR

[4] U. Einmahl, “Law of the iterated logarithm type results for random vectors with infinite second moment”, Matematica Applicanda, 44:1 (2016), 167–181 | MR

[5] U. Einmahl, D. Li, “Some results on two-sided LIL behavior”, Ann. Probab., 33:4 (2005), 1601–1624 | DOI | MR

[6] X. Fernique, “Régularité de processus gaussiens”, Inventiones Mathematicae, 12 (1971), 304–320 | DOI | MR

[7] V. Goodman, “Characteristics of normal samples”, Ann. Probab., 16:3 (1988), 1281–1290 | DOI | MR

[8] M. Ledoux, M. Talagrand, Probability in Banach Spaces, Springer, 1991 | MR

[9] S. N. Majumdar, A. Comptet, J. Randon-Furling, “Random convex hulls and extreme value statistics”, J. Stat. Phys., 138 (2010), 955–1009 | DOI | MR

[10] V. V. Petrov, Limit Theorems of Probability Theory. Sequences of Independent Random Variables, Clarendon Press, Oxford, 1995 | MR

[11] M. Talagrand, “Sur l'integrabilité des vecteurs gaussiens”, Z. Wahrscheinlich. Verw. Geb., 68 (1984), 1–8 | DOI | MR