Integral representations and convergence of the renewal density

We derive integral representations for the renewal density u associated with a square integrable probability density p on [0,∞) having finite expected value μ. 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 limt→∞(u(t)−p(t))=1/μ.