Varadhan Functions, Variances, and Means on Compact Riemannian Manifolds

TL;DR

The paper develops a Varadhan-based framework to study smooth approximations of Fréchet statistics on compact Riemannian manifolds by introducing $t$-Varadhan functions, variances, and means via the logarithmic heat kernel. It establishes uniform LLNs and CLTs for empirical versions of these statistics at fixed times, and links the $t\to0^+$ limit to Fréchet-mean theory through small-time gradient and Hessian analyses, including a nontrivial correction term $J_\mu$ arising from the cut locus. The results generalize Fréchet-mean CLTs to settings where standard smoothness fails and show how the diffusion-like $t$-means converge to Fréchet means under mild regularity, with explicit connections to recent work on non-standard asymptotics. The circle and torus examples demonstrate the explicit form of the correction term and its agreement with known Fréchet-mean corrections, suggesting broad applicability of Varadhan-type statistics for manifold-valued data.

Abstract

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.

Theorems & Definitions (42)

• Theorem : varadhan1967behavior
• Definition 2.1
• Proposition 3.1
• Remark 3.2
• Theorem 3.3
• Theorem 3.4
• Remark 3.5
• Theorem 4.1
• Theorem 4.2
• Remark 4.3