For $1 le p < infty$, the Frechet $p$-mean of a probability distribution $mu$ on a metric space $(X,d)$ is the set $F_p(mu) := {arg,min}_{xin X}int_{X}d^p(x,y), dmu(y)$, 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 Frechet $p$-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 $n to infty$.
Download