For $1\le p \le \infty$, the Fr\'echet $p$-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 Fr\'echet 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.

翻译：对于 $1\le p \le \infty$，度量空间上概率测度的Fréchet $p$-均值是集中趋势的重要概念，它推广了实数线上均值($p=2$)和中位数($p=1$)的通常定义。本文证明了关于Fréchet均值及相关对象的一族极限定理，这些对象一般情况下构成随机闭集序列。一方面，我们表明许多极限定理（强大数定律、遍历定理和大偏差原理）可通过纯拓扑学考虑，从概率测度空间上的相应定理简单导出。另一方面，我们给出了在 $T_2$ 拓扑（特别是Fell拓扑）下成立强大数定律的首批充分条件，并证明该条件在某些特殊情形下是必要的。此外，我们还讨论了本文结果在统计学与计算科学中的意义。