Reference needed: Donsker's Invariance Principle for Riemannian Manifolds After an extensive unsuccessful search: I need a reference (preferably a book) for the Donsker's invariance principle for Riemannian manifolds. Thanks.
 A: The generalization of Donsker's theorem from $N$-dimensional Euclidean space to general Riemannian manifolds has been worked out by Erik Jørgensen, The Central Limit Problem for Geodesic Random Walks.

The purpose of the present work is to
  consider the problem of defining the
  concept of a random walk in a general
  Riemannian manifold ${\cal M}$, and to
  investigate the behavior in the limit
  of a sequence of such random walks. It
  will be shown that such a sequence,
  under reasonable assumptions,
  converges to a diffusion process in
  ${\cal M}$, and in particular Brownian motion
  processes will be obtained as limits
  of sequences of random walks with
  identically distributed steps. The
  results which we arrive at in this
  paper are general versions of
  well-known classical results
  concerning the transition from random
  walks to diffusion processes, for
  instance: the central limit theorem
  and Donsker's theorem.

A: D. Khoshnevisan, Lecture notes on Donsker's theorem
http://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf
A: You will probably interested by the paper 
Empirical invariance principle for ergodic torus automorphisms; Genericity. Olivier Durieu, Philippe Jouan. Stochastics and Dynamics, volume 8, 2, p. 173-195, 2008. 
It is available here.
