Large deviation principles for renewal-reward processes

TL;DR

This paper proves a precise large deviation principle for renewal-reward processes in Banach spaces, even without exponential moment assumptions, and provides conditions for exponential tightness to establish a full principle.

Contribution

It introduces a sharp large deviation principle for renewal-reward processes in Banach spaces without requiring exponential moment conditions, extending existing results.

Findings

Established a weak large deviation principle without exponential moment assumptions.

Provided sufficient conditions for exponential tightness of renewal-reward processes.

Demonstrated a full large deviation principle under these conditions.

Abstract

We establish a sharp large deviation principle for renewal-reward processes, supposing that each renewal involves a broad-sense reward taking values in a real separable Banach space. In fact, we demonstrate a weak large deviation principle without assuming any exponential moment condition on the law of waiting times and rewards by resorting to a sharp version of Cram\'er's theorem. We also exhibit sufficient conditions for exponential tightness of renewal-reward processes, which leads to a full large deviation principle.