Some results for the delayed renewal function

In this paper, the delayed or modified renewal function is examined in some detail. Some properties, a characterization and an improved asymptotic result are discussed. Also, we obtain some “general” upper and lower bounds for the delayed renewal function in the sense that the bounds are valid for any arbitrary distributions of the inter-arrival times, as well as bounds when the distributions of the interarrival times belong to some non-parametric families of distributions, satisfying an aging property. Some bounds for the delayed renewal function which are based on stochastic comparisons of the interarrival times, are also given.