论文标题
有关高斯凸壳的收敛的更多信息
More on the convergence of Gaussian convex hulls
论文作者
论文摘要
连续凸壳的“大数字”，用于弱依赖的高斯序列$ \ {x_n \} $具有相同边缘分布的“ X_n \} $，则将序列$ \ {x_n \} $具有弱极限时，将其扩展到情况。令$ \ mathbb {b} $为具有共轭空间$ \ mathbb {b}^\ ast $的可分开的Banach空间。令$ \ {x_n \} $为中心的$ \ mathbb {b} $ - 值满足两个条件的高斯序列：1）$ x_n \ rightarrow x \; \; \; \; $; $; $ and 2）in \ in \ in \ in \ in \ in \ mathbb {b} \ infty} e \ langle x_n，x^*\ rangle \ langle x_m，x^*\ rangle \; \; = \; \; 0。 $$，然后使用概率1，归一化的凸壳$$ w_n = \ frac {1} {（2 \ ln n）^{1/2}}} \，{\ rm cons} \ {\ {\，x_1，x_1，\ ldots，\ ldots，x_ {n} \，x {n} \，\，\，\，\，\，\，\ implip， Gaussian $ \ Mathbb {B} $ - 有价值的随机元素$X。$此外，还讨论了一些相关问题。
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 $$ W_n = \frac{1}{(2\ln n)^{1/2}}\,{\rm conv} \{\,X_1,\ldots,X_{n}\,\} $$ 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.
