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For $1\le p \le \infty$, the Fr\'echet $p$-mean of a probability measure on ametric space is an important notion of central tendency that generalizes theusual notions in the real line of mean ($p=2$) and median ($p=1$). In this workwe prove a collection of limit theorems for Fr\'echet means and relatedobjects, which, in general, constitute a sequence of random closed sets. On theone hand, we show that many limit theorems (a strong law of large numbers, anergodic theorem, and a large deviations principle) can be simply descended fromanalogous theorems on the space of probability measures via purely topologicalconsiderations. On the other hand, we provide the first sufficient conditionsfor 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 somespecial cases. We also discuss statistical and computational implications ofthe results herein.

Limit Theorems for Fr\'echet Mean Sets