# Large Deviations in Discrete-Time Renewal Theory

## TL;DR

This paper develops precise large deviation principles for cumulative rewards in a discrete-time renewal model, including the polymer pinning model, using convexity and super-additivity techniques.

## Contribution

It establishes sharp large deviation results for a broad class of rewards in renewal processes, extending to the polymer pinning model with a novel approach.

## Key findings

- Large deviation principles for rewards in renewal models
- Extension to the polymer pinning model
- Method based on convexity and super-additivity

## Abstract

We establish sharp large deviation principles for cumulative rewards associated with a discrete-time renewal model, supposing that each renewal involves a broad-sense reward taking values in a real separable Banach space. The framework we consider is the pinning model of polymers, which amounts to a Gibbs change of measure of a classical renewal process and includes it as a special case. We first tackle the problem in a constrained pinning model, where one of the renewals occurs at a given time, by an argument based on convexity and super-additivity. We then transfer the results to the original pinning model by resorting to conditioning.