Quenched large deviations for randomly weighted geodesic random walks

TL;DR

This paper establishes a universal large deviation principle for weighted geodesic random walks on Riemannian manifolds, extending previous real-valued results to more general settings and vector spaces.

Contribution

It generalizes large deviation results for geodesic random walks to weighted cases on manifolds and vector spaces, broadening the scope of prior work.

Findings

Weighted geodesic random walks satisfy a large deviation principle.

The results extend previous real-valued cases to general vector spaces.

Methodology connects large deviations in tangent spaces to those on manifolds.

Abstract

We consider weighted geodesic random walks in a complete Riemannian manifold $(M,g)$. We show that for almost all sequences of weights (with respect to a suitable measure), these weighted geodesic random walks satisfy, when suitably scaled, a large deviation principle with a universal rate function. This extends the results from [3], where this was shown for the real-valued case. It turns out the argument is also valid for general vector spaces. This allows us to use the methodology of [9], in which large deviations for geodesic random walks are obtained from large deviation estimates for associated random walks in tangent spaces.