An omnibus CLT for Fréchet means and nonparametric inference on non-Euclidean spaces

We derive a general central limit theorem from which many of the existing CLTs for Fr\'echet means follow immediately. It also improves upon an earlier CLT for intrinsic sample means on Riemannian manifolds due to Bhattacharya and Patrangenaru (2005). Applications are provided to some stratified spaces of recent interest, and to the space of symmetric positive definite matrices arising in diffusion tensor imaging.