论文标题
Fréchet均值集的大量法律
Strong Laws of Large Numbers for Generalizations of Fréchet Mean Sets
论文作者
论文摘要
一个随机变量$ y $的fréchet平均值$ y $在公制空间$中（\ Mathcal Q，d）$是公制空间的元素，该元素将$ q \ q \ mapsto \ mathbb e [d（y，q）^2] $最小化。这种最小化可能不是唯一的。我们研究了一组广义的Fréchet手段的大量法律。考虑以后的概括：$ \ mathbb e [d（y，q）^α] $ for $α> 0 $，$ \ mathbb e [h（d（y，q）] $的最小化器，用于$ h $ for Mathbb e [y mathbb e [y Mathbb e [y Mathbb e [y Mathbb e [y Mathbb e [y Mathbb e]的$ h $ h $，功能$ \ mathfrak c $。我们在外部极限和单侧Hausdorff距离内显示了这些集合的经验版本的收敛性。得出的结果仅需要最小的假设。
A Fréchet mean of a random variable $Y$ with values in a metric space $(\mathcal Q, d)$ is an element of the metric space that minimizes $q \mapsto \mathbb E[d(Y,q)^2]$. This minimizer may be non-unique. We study strong laws of large numbers for sets of generalized Fréchet means. Following generalizations are considered: the minimizers of $\mathbb E[d(Y, q)^α]$ for $α> 0$, the minimizers of $\mathbb E[H(d(Y, q))]$ for integrals $H$ of non-decreasing functions, and the minimizers of $\mathbb E[\mathfrak c(Y, q)]$ for a quite unrestricted class of cost functions $\mathfrak c$. We show convergence of empirical versions of these sets in outer limit and in one-sided Hausdorff distance. The derived results require only minimal assumptions.
