We present some partial results on quantitative exponential convergence estimates for the renewal theorem when the inter arrival distribution has a uniform component and one finite exponential monent. We improve to that end the classic idea of coupling the age processes of a delayed renewal process and a non delayed one in order to get explicit estimates. A key element is our construcion of a coalescent coupling between two copies of the process that succeeds with at least some explicit probability which is independent of their initial relative delay. This construction and the computation of the lower bound rely on the use of the uniform component to couple inter arrival lengths with small enough «shifted» of those random variables. A second element is the study of a biased random walk associated with two independent copies of the process, which allows us to obtain, via Lyapunov and martingale techniques, an estimate of the total elapsed time required to observe renewals of the two copies within a time-interval of (large enough) prescribed length. Based on joint work in progress with Jean-Baptiste Bardet and Alejandra Christen.

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