Bounds on the error in the linear approximation to the renewal function

It is assumed throughout that we have a renewal situation, specified by a ‘lifetime’ probability density f (x) which decreases in an exponential manner as x→∞. In particular, if f(x) is dominated by the exponential function Ae−βx for some A > 0, β > 0, it is shown that the error in the linear approximation to the renewal function also decays exponentially fast as x→∞. To determine the constants involved, it is necessary to have some knowledge about the location of the roots of the characteristic equation. One method for obtaining the required information is described, and applications given to the case of gamma density functions. A similar analysis is possible (and the details are indicated) for approximations to the renewal density, and to higher renewal moments.