Central Limit Theorems for Sample Fr\'echet Means of Manifold-Valued Markov Chains

Abstract

In this article, we establish central limit theorems for sample Fr\'echet means of stationary ergodic Markov chains taking values in manifolds, extending the asymptotic theory previously developed for independent observations to a class of dependent manifold-valued processes. Our results derive the asymptotic normality of the sample Fr\'echet mean from a central limit condition at the population Fr\'echet mean, under suitable local regularity conditions. We further provide sufficient geometric and probabilistic conditions under which these assumptions hold, formulated in terms of curvature bounds and a Wasserstein mixing condition. As an application, we establish a central limit theorem for sample Fr\'echet means for a class of random dynamical systems generated by contractive random maps.