Central Limit Theorems on Compact Metric Spaces

Steven Rosenberg, Jie Xu

Published 2020-01-09Version 1

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 space. However, the sample mean makes no sense back on $K$, so using $\eta$ we develop a CLT for the sample Fr\'echet mean. 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 then obtain an $L^2$-CLT for both the sample mean and the sample Fr\'echet mean. Since the $L^2$ and $L^\infty$ norms play important roles, in the last section we develop a metric-measure criterion relating $d$ and $\eta$ under which all $L^p$ norms are equivalent.