Quantitative exponential bounds for the renewal theorem with spread-out distributions

TL;DR

This paper provides explicit exponential convergence bounds for renewal theorems with spread-out distributions, using coupling methods and properties of Laplace transforms, advancing quantitative understanding of renewal processes.

Contribution

It introduces a new approach to quantify exponential convergence in renewal theorems with spread-out distributions through explicit bounds and coupling techniques.

Findings

Established explicit exponential convergence estimates for renewal theorems.

Linked convergence rates to properties of the inter-arrival distribution and Laplace transforms.

Utilized coupling and Lyapunov-Doeblin methods for the proofs.

Abstract

We establish explicit exponential convergence estimates for the renewal theorem, in terms of a uniform component of the inter arrival distribution, of its Laplace transform which is assumed finite on a positive interval, and of the Laplace transform of some related random variable. Our proof is based on a coupling construction relying on discrete-time Markovian structures that underly the renewal processes and on Lyapunov-Doeblin type arguments.