From uniform renewal theorem to uniform large and moderate deviations for renewal-reward processes

TL;DR

This paper develops a uniform renewal theorem and establishes uniform large and moderate deviations principles for renewal-reward processes, broadening the understanding of their probabilistic behavior under weaker conditions.

Contribution

It introduces a uniform renewal theorem derived from Blackwell's theorem and proves uniform LDP and MDP for renewal-reward processes with weaker moment conditions.

Findings

Established a uniform renewal theorem from Blackwell's renewal theorem.

Proved a uniform large deviations principle for renewal-reward processes.

Derived a moderate deviations principle under weaker conditions than exponential moments.

Abstract

A uniform key renewal theorem is deduced from the uniform Blackwell's renewal theorem. A uniform LDP (large deviations principle) for renewal-reward processes is obtained, and MDP (moderate deviations principle) is deduced under conditions much weaker than existence of exponential moments.