Quantitative Exponential Bounds for the Renewal Theorem with Spread-Out Distributions

Abstract

We establish 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. Although our bounds are not sharp, our approach provides tractable constructive estimates for the renewal theorem which are computable (theoretically and numerically, at least) for a general class of inter-arrival distributions. The proof uses a coupling, and relies on Lyapunov - Doeblin type arguments for some discrete time regenerative structure, which we associate with the renewal processes.