Limit theorems for empirical Fréchet means of independent\ud and non-identically distributed manifold-valued\ud random variables

摘要

We prove weak laws of large numbers and central limit theorems\udof Lindeberg type for empirical centres of mass (empirical Fréchet means)\udof independent nonidentically distributed random variables taking values in\udRiemannian manifolds. In order to prove these theorems we describe and\udprove a simple kind of Lindeberg–Feller central approximation theorem for\udvector-valued random variables, which may be of independent interest and\udis therefore the subject of a self-contained section. This vector-valued result\udallows us to clarify the number of conditions required for the central limit\udtheorem for empirical Fréchet means, while extending its scope.