Fréchet均值集合的极限定理
Limit Theorems for Fréchet Mean Sets
AI总结：
本文研究度量空间上Fréchet均值集合的极限定理，通过拓扑方法从概率测度空间上的定理推导出强大数定律、遍历定理和大偏差原理，并首次给出$T_2$拓扑下强大数定律成立的充分条件，同时讨论其在统计与计算中的意义。
摘要：
对于 $1\le p \le \infty$，度量空间上概率测度的Fréchet $p$-均值是中心趋势的一个重要概念，它推广了实直线上通常的均值（$p=2$）和中位数（$p=1$）概念。在本文中，我们证明了关于Fréchet均值及相关对象的一系列极限定理，这些对象通常构成随机闭集序列。一方面，我们证明了许多极限定理（强大数定律、遍历定理和大偏差原理）可以通过纯拓扑考虑，从概率测度空间上的类似定理简单地推导出来。另一方面，我们首次给出了在$T_2$拓扑（特别是Fell拓扑）下强大数定律成立的充分条件，并证明该条件在某些特殊情形下是必要的。我们还讨论了本文结果在统计和计算方面的意义。
英文摘要：
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.
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