Sample Path Large Deviations for Random Walks on Regular Trees

TL;DR

This paper establishes a large deviation principle for the sample paths of nearest-neighbor random walks on regular trees, providing a convex rate function and an explicit expression via the Fenchel-Legendre transform.

Contribution

It introduces the first large deviation principle for sample paths of random walks on regular trees, including an explicit rate function expression.

Findings

Proves the sample path large deviation principle for regular tree random walks.

Derives an explicit rate function using Fenchel-Legendre transform.

Provides a convex rate function for the deviation probabilities.

Abstract

This paper investigates the large deviation problem in the sample path space of the nearest-neighbor random walks on regular trees. We establish the sample path large deviation principle for the law of the distance from a nearest random walk on a regular tree to the root with a good convex rate function. Furthermore, we derive an implicit expression for the rate function via the Fenchel-Legendre transform of the log-moment generating function.