When the scalar quadratic variation of a martingale on a Riemannian manifold is finite almost surely, then the martingale converges almost surely in the one-point compactification of the manifold. A partial converse due to Zheng Wei-an is also proved. No curvature conditions on the manifold are required.
Cite this article
Richard W.R. Darling, Convergence of Martingales on a Riemannlan Manifold. Publ. Res. Inst. Math. Sci. 19 (1983), no. 2, pp. 753–763