Motivated by Varadhan's theorem, we introduce Varadhan functions, variances, and means on compact Riemannian manifolds as smooth approximations to their Fréchet 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 Fréchet means, without assumptions on the geometry of the cut locus.

翻译：受Varadhan定理启发，我们在紧致黎曼流形上引入Varadhan函数、方差与均值，作为其Fréchet对应量的光滑逼近。给定独立同分布样本，我们证明了其经验版本的一致大数定律。进一步地，我们证明了对于任意固定$t\ge0$的Varadhan函数与方差、以及对于任意固定$t>0$的Varadhan均值的中心极限定理。通过研究Varadhan函数梯度与Hessian矩阵的短时渐近性质，我们在无需割迹几何假设的条件下，建立了与Fréchet均值中心极限定理的深刻联系。