Central Limit Theorems on Compact Metric Spaces

Abstract

We produce a series of Central Limit Theorems (CLTs) associated to compact metric measure spaces $(K,d,\eta)$, with $\eta$ a reasonable probability measure. For the first CLT, we can ignore $\eta$ by isometrically embedding $K$ into ${\mathcal C}(K)$, the space of continuous functions on $K$ with the sup norm, and then applying known CLTs for sample means on Banach spaces (Theorem 3.1). However, the sample mean makes no sense back on $K$, so using $\eta$ we develop a CLT for the sample Fr\'echet mean (Corollary 4.1). This involves working on the closed convex hull of the embedded image of $K$. To work in the easier Hilbert space setting of $L^2(K,\eta)$, we have to modify the metric $d$ to a related metric $d_\eta$. We obtain an $L^2$-CLT for both the sample mean and the sample Fr\'echet mean (Theorem 5.1), and we relate the Fr\'echet sample and population means on the closed convex hull to the Fr\'echet means on the image of $K$. Since the $L^2$ and $L^\infty$ norms play important roles, in Section 6 we develop a metric-measure criterion relating $d$ and $\eta$ under which all $L^p$ norms are equivalent.