Varadhan Functions, Variances, and Means on Compact Riemannian Manifolds

Motivated by Varadhan's theorem, we introduce Varadhan functions, variances, and means on compact Riemannian manifolds as smooth approximations to their Fr\'echet counterparts. Given independent and identically distributed samples, we prove uniform laws of large numbers for their empirical versions. Furthermore, we prove central limit theorems for Varadhan functions and variances for each fixed $t\ge0$, and for Varadhan means for each fixed $t>0$. By studying small time asymptotics of gradients and Hessians of Varadhan functions, we build a strong connection to the central limit theorem for Fr\'echet means, without assumptions on the geometry of the cut locus.