Random walks on groups have been utilized to study a wide array of mathematics, e.g. number theory, the spectral theory of Schrodinger operators, and homogeneous dynamics. Under sufficiently nice dynamical assumptions, these random walks obey central limit theorems. We will discuss some joint work with Omar Hurtado in which we introduce a natural family of topologies on spaces of probability measures, and study continuity and stability of statistical properties of random walks on linear groups over local fields. We are able to extend large deviation results known in the Archimedean case to non-Archimedean local fields and also demonstrate certain large deviation estimates for heavy tailed distributions unknown even in the Archimedean case. Time permitting, we will discuss applications to Schodinger operators (an Anderson localization result) and hyperbolic geometry (a stable geodesic counting result).