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