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Thank you, Stephan, for correcting me.

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.

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.

• Probability: Theory and Examples, R. Durrett (2010), chapter 8.

• A course in probability theory, K.L. Chung (2000).

From a review of Chung's book:

There is a substantial chapter on invariance theorems, including Donsker's invariance principle, the Doob-Donsker proof of the Kolmogorov-Smirnov theorem and finally a general invariance principle of Skorokhod, the proof of which is only sketched.

• Convergence of Probability Measures, by P. Billingsley (1999)

From a review of Billingsley's book:

This book is about weak- convergence methods in metric spaces, with applications sufficient to show their power and utility. The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued.

Thank you, Stephan, for correcting me.

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.

Probability: Theory and Examples, R. Durrett (2010), chapter 8.

A course in probability theory, K.L. Chung (2000).

From a review of Chung's book:

There is a substantial chapter on invariance theorems, including Donsker's invariance principle, the Doob-Donsker proof of the Kolmogorov-Smirnov theorem and finally a general invariance principle of Skorokhod, the proof of which is only sketched.

for Donsker's theorem in the context of more general metric spaces, see:

Convergence of Probability Measures, by P. Billingsley (1999)

• Probability: Theory and Examples, R. Durrett (2010), chapter 8.

• A course in probability theory, K.L. Chung (2000).

From a review of Chung's book:

There is a substantial chapter on invariance theorems, including Donsker's invariance principle, the Doob-Donsker proof of the Kolmogorov-Smirnov theorem and finally a general invariance principle of Skorokhod, the proof of which is only sketched.

• Convergence of Probability Measures, by P. Billingsley (1999)

From a review of Billingsley's book:

This book is about weak- convergence methods in metric spaces, with applications sufficient to show their power and utility. The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued.