论文标题
限制fréchet平均套件的定理
Limit Theorems for Fréchet Mean Sets
论文作者
论文摘要
对于$ 1 \ le p \ le \ infty $，fréchet$ p $ - 公制空间上的概率度量是一个重要的中心趋势的概念，它在平均值（$ p = 2 $）和中间（$ p = 1 $）中概括了通常的概念（$ p = 2 $）。在这项工作中，我们证明了Fréchet手段和相关对象的限制定理的集合，通常，该对象构成了一系列随机封闭集。一方面，我们表明许多限制了定理（大量的强大定律，千古定理和较大的偏差原理）可以简单地从类似的定理中降到概率措施的类似定理中，这是通过纯粹的拓扑考虑因素来的。另一方面，我们提供了第一个足够的条件，以使大量法律以$ T_2 $拓扑（尤其是秋季拓扑结构）保留，并且在某些特殊情况下，我们表明这种情况是必要的。我们还讨论了本文结果的统计和计算含义。
For $1\le p \le \infty$, the Fréchet $p$-mean of a probability measure on a metric space is an important notion of central tendency that generalizes the usual notions in the real line of mean ($p=2$) and median ($p=1$). In this work we prove a collection of limit theorems for Fréchet means and related objects, which, in general, constitute a sequence of random closed sets. On the one hand, we show that many limit theorems (a strong law of large numbers, an ergodic theorem, and a large deviations principle) can be simply descended from analogous theorems on the space of probability measures via purely topological considerations. On the other hand, we provide the first sufficient conditions for the strong law of large numbers to hold in a $T_2$ topology (in particular, the Fell topology), and we show that this condition is necessary in some special cases. We also discuss statistical and computational implications of the results herein.
