Integral representations and convergence of the renewal density

Abstract

We derive integral representations for the renewal density u associated with a square integrable probability density p on [0, infinity) having finite expected value mu. These representations express u in terms of the real and the imaginary part of the Fourier transform of p, considered as a function on the lower complex half plane. We use them to give simple global integrability conditions on p under which lim(t ->infinity) (u(t) - p(t)) = 1/mu. (c) 2005 Elsevier Inc. All rights reserved.