We produce a series of Central Limit Theorems (CLTs) associated to compact metric measure spaces $(K,d,\eta )$. The main obstacle is the impossibility of averaging $K$-valued random variables. This is overcome by using isometric images of $K$ inside a Banach space or a Hilbert space, after which we can apply results for CLTs on these spaces.
"Central limit theorems on compact metric spaces." Electron. Commun. Probab. 25 1 - 10, 2020. https://doi.org/10.1214/20-ECP336