Large deviations for random walks on Lie groups

TL;DR

This paper establishes large deviation principles for random walks on Lie groups, utilizing a rescaling approach that simplifies analysis without dilations, and applies results to stochastic matrices.

Contribution

It introduces a new method for analyzing large deviations in Lie groups by rescaling increments, avoiding complex dilations, and extends the theory to stochastic matrices.

Findings

Large deviation principles are proven for random walks on Lie groups.

The rescaling approach simplifies the analysis of non-commutative structures.

Application to stochastic matrices demonstrates practical relevance.

Abstract

We study large deviations for random walks on Lie groups defined by $\sigma_n^n = \exp(\frac1nX_1)\cdots\exp(\frac1nX_n)$, where $\{X_n\}_{n\geq1}$ is an i.i.d sequence of bounded random variables in the Lie algebra $\mathfrak{g}$. We follow a similar approach as in the proof of large deviations for geodesic random walks as given in [Ver19]. This approach makes it possible to simply rescale the increments of the random walk, without having to resort to dilations in order to reduce the influence of higher order commutators. Finally, we will apply this large deviation result to the Lie group of stochastic matrices.