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After an extensive unsuccessful search: I need a reference (preferably a book) for the Donsker's invariance principle for Riemannian manifolds. Thanks.

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up vote 4 down vote accepted

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.

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@Carlo So far it looks very promising - and provides pretty much interesting reading material ... Thanks a lot! @Stephan Thank you very much for your helping hand. –  kassandra Apr 3 '13 at 18:49
    
yes, thank you Stephan for your expert advise. –  Carlo Beenakker Apr 3 '13 at 19:46
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D. Khoshnevisan, Lecture notes on Donsker's theorem

http://www.math.utah.edu/~davar/ps-pdf-files/donsker.pdf

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Great lecture notes on Donsker's theorem in $\mathbb{R}$. However, I do not see how this could be applied to a generic manifold setting. –  Stephan Sturm Apr 3 '13 at 13:29
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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.

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