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2019 Large deviations for geodesic random walks
Rik Versendaal
Electron. J. Probab. 24: 1-39 (2019). DOI: 10.1214/19-EJP351


We provide a direct proof of Cramér’s theorem for geodesic random walks in a complete Riemannian manifold $(M,g)$. We show how to exploit the vector space structure of the tangent spaces to study large deviation properties of geodesic random walks in $M$. Furthermore, we reveal the geometric obstructions one runs into.

To overcome these obstructions, we provide a Taylor expansion of the inverse Riemannian exponential map, together with appropriate bounds. Furthermore, we compare the differential of the Riemannian exponential map to parallel transport. Finally, we show how far geodesics, possibly starting in different points, may spread in a given amount of time.

With all geometric results in place, we obtain the analogue of Cramér’s theorem for geodesic random walks by showing that the curvature terms arising in this geometric analysis can be controlled and are negligible on an exponential scale.


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Rik Versendaal. "Large deviations for geodesic random walks." Electron. J. Probab. 24 1 - 39, 2019.


Received: 13 December 2018; Accepted: 5 August 2019; Published: 2019
First available in Project Euclid: 13 September 2019

zbMATH: 07107400
MathSciNet: MR4003146
Digital Object Identifier: 10.1214/19-EJP351

Primary: 58C99, 60F10, 60G50


Vol.24 • 2019
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