Large deviations for irreducible random walks on relatively hyperbolic groups

TL;DR

This paper establishes large deviation principles for irreducible random walks on relatively hyperbolic groups, providing a detailed rate function description under certain moment conditions.

Contribution

It proves the existence of weak and full large deviation principles for these random walks, with explicit rate functions related to logarithmic moment generating functions.

Findings

Weak large deviation principle with convex rate function

Full large deviation principle under exponential moment finiteness

Rate function as Fenchel-Legendre transform of the logarithmic moment generating function

Abstract

We show existence of the weak large deviation principle, with a convex rate function, for the renormalized distance from the starting point of irreducible random walks on relatively hyperbolic groups. Under the assumption of finiteness of exponential moments, the full large deviation principle holds, and the rate function governing it can be expressed as the Fenchel-Legendre transform of the limiting logarithmic moment generating function of the sequence of renormalized distances.