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.
UDC:
519.2 Received: 25.07.2022
Language: English