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Strong Law of Large Numbers for Measures of Central Tendency and Dispersion of Random Variables in Compact Metric Spaces
Given a sample of independent random variables $Z_1, Z_2, \cdots, Z_n$ with identical distribution $p$ on a compact metric space $(M, d)$, a measure of central tendency is a sample centroid (of order $r > 0$) defined as a point $\hat{X}_n$ in $M$ satisfying $\frac{1}{n} \sum^n_{i=1} d^r(\hat{X}_n, Z_i) = \inf_{x …