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.
Highlights
Random walks are among the most extensively studied discrete stochastic processes
It turns out that it is possible to only study the underlying geometry in order to prove Cramér’s theorem. This gives us new insight in what geometrical aspects allow us to still obtain the large deviation principle for rescaled geodesic random walks, even though the geodesic random walk is in general no longer a simple function of its increments
Our method of proving the large deviations for Zn does not immediately allow us to conclude that Zn and Zn are exponentially equivalent, the main idea of our proof does rely on the fact that we can relate and compare the geodesic random walk to a sum of independent, identically distributed random variables in the tangent space at x0, following the distribution μx0
Summary
Random walks are among the most extensively studied discrete stochastic processes. Given a sequence of random variables {Xn}n≥1 in some vector space V , one defines the random walk with increments {Xn}n≥1 as the random variable n. It turns out that it is possible to only study the underlying geometry in order to prove Cramér’s theorem This gives us new insight in what geometrical aspects allow us to still obtain the large deviation principle for rescaled geodesic random walks, even though the geodesic random walk is in general no longer a simple function of its increments. Our method of proving the large deviations for Zn does not immediately allow us to conclude that Zn and Zn are exponentially equivalent, the main idea of our proof does rely on the fact that we can (in some sense) relate and compare the geodesic random walk to a sum of independent, identically distributed random variables in the tangent space at x0, following the distribution μx0.
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