Large deviations for geodesic random walks

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Abstract

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.

Original languageEnglish
Article number93
Pages (from-to)1-39
Number of pages39
JournalElectronic Journal of Probability
Volume24
DOIs
Publication statusPublished - 2019

Keywords

  • Cramér’s theorem
  • Geodesic random walks
  • Jacobi fields
  • Large deviations
  • Riemannian exponential map
  • Spreading of geodesics

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