  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.
