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 Z₁, Z₂, ⋯, Zₙ 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 X̂ₙ in M satisfying 1/n ∑ⁿᵢ₌₁ dʳ(X̂ₙ, Zᵢ) = infx ∈ M 1/n ∑ⁿᵢ₌₁ dʳ(x, Zᵢ). A (population) centroid of Z is any point x^ in M such that ∫M dʳ(x^, z) dp(z) = infx ∈ M ∫M dʳ(x, z) dp(z). The quantity 1/n ∑ⁿᵢ₌₁ dʳ(X̂ₙ, Zᵢ) itself is called the sample variation, whereas ∫M dʳ(x^, z) dp(z) is the variation of Z. This paper establishes almost sure convergence for the sample centroid and variation to the corresponding population values for all orders $r > 0$. Convergence is also proved for the case when the sample centroid is restricted to be one of the sample values.