Strong laws of large numbers for Fréchet means

Steven N. Evans, Adam Q. Jaffe

Published 2020-12-23Version 1

For $1 \le p < \infty$, the Fr\'echet $p$-mean of a probability distribution $\mu$ on a metric space $(X,d)$ is the set $F_p(\mu) := {\arg\,\min}_{x\in X}\int_{X}d^p(x,y)\, d\mu(y)$, which is taken to be empty if no minimizer exists. Given a sequence $(Y_i)_{i \in \mathbb{N}}$ of independent, identically distributed random samples from some probability measure $\mu$ on $X$, the Fr\'echet $p$-means of the empirical measures, $F_p(\frac{1}{n}\sum_{i=1}^{n}\delta_{Y_i})$ form a sequence of random closed subsets of $X$. We investigate the senses in which this sequence of random closed sets and related objects converge almost surely as $n \to \infty$.