More on the convergence of Gaussian convex hulls

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 $x^{\ast }\in {\mathbb{B}}^{\ast }$ $$\underset{n,m,|n-m|\to \infty }{\lim }E⟨X_n,x^{\ast }⟩⟨X_m,x^{\ast }⟩=0.$$ Then with probability 1 the normalized convex hulls $$W_n=\frac{1}{(2\ln n{)}^{1/2}}\mathrm{conv}\{X_1,\dots ,X_n\}$$ 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.

12 pages

References in corpus (1)

• Some results on two-sided LIL behavior