Large deviations for geodesic random walks. (English) Zbl 1466.60064
Summary: 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.
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.
MSC:
| 60F10 | Large deviations |
| 60G50 | Sums of independent random variables; random walks |
| 58C99 | Calculus on manifolds; nonlinear operators |
Keywords:
large deviations; Cramér’s theorem; geodesic random walks; Riemannian exponential map; Jacobi fields; spreading of geodesicsReferences:
| [1] | Amir Dembo and Ofer Zeitouni, Large deviations techniques and applications, second ed., Applications of Mathematics (New York), vol. 38, Springer-Verlag, New York, 1998. · Zbl 0896.60013 |
| [2] | Jean-Dominique Deuschel and Daniel W. Stroock, Large deviations, Pure and Applied Mathematics, vol. 137, Academic Press, Inc., Boston, MA, 1989. · Zbl 0705.60029 |
| [3] | Jin Feng and Thomas G. Kurtz, Large deviations for stochastic processes, Mathematical Surveys and Monographs, vol. 131, American Mathematical Society, 2006. · Zbl 1113.60002 |
| [4] | Theodore Frankel, The geometry of physics, second ed., Cambridge University Press, Cambridge, 2004, An introduction. · Zbl 1049.58001 |
| [5] | Frank den Hollander, Large deviations, Fields Institute Monographs, vol. 14, American Mathematical Society, Providence, RI, 2000. · Zbl 0949.60001 |
| [6] | Erik Jørgensen, The central limit problem for geodesic random walks, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32 (1975), 1-64. · Zbl 0292.60103 |
| [7] | Wilhelm Klingenberg, Riemannian geometry, de Gruyter Studies in Mathematics, vol. 1, Walter de Gruyter & Co., Berlin-New York, 1982. · Zbl 0495.53036 |
| [8] | Richard Kraaij, Frank Redig, and Rik Versendaal, Classical large deviations theorems on complete riemannian manifolds, to appear in Stochastic Processes and their Applications, 2018. · Zbl 1448.60065 |
| [9] | John M. Lee, Riemannian manifolds: An introduction to curvature, Graduate Texts in Mathematics, vol. 176, Springer-Verlag, New York, 1997. · Zbl 0905.53001 |
| [10] | R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. · Zbl 0193.18401 |
| [11] | Michael Spivak, A comprehensive introduction to differential geometry. Vol. I, second ed., Publish or Perish, Inc., Wilmington, Del., 1979. · Zbl 0439.53003 |
| [12] | D. W. Stroock, An introduction to the theory of large deviations, Universitext, Springer-Verlag, New York, 1984. · Zbl 0552.60022 |
| [13] | Stefan Waldmann, Geometric wave equations, Lecture notes; ArXiv:1208.4706v1 (2012), no. 1208.4706v1. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
