Approximations to generalized renewal measures

Let {Zj, j⩾1} be a sequence of nonnegative continuous random variables. Given an arbitrary function g : [0,∞)→[0,∞), a renewal function associated with this sequence is defined asS(b)=∑j=1∞ g(j)P{Zj<b}, b>0.Due to possible complexity of calculating the probabilities P{Zj<b}, computation of S(b) is often intractable. er a sequence of positive numbers {mj, j⩾1} and defineS∗(b)=∑j=1∞ g(j)I{mj<b}.Clearly, S∗(b) is much easier to calculate than S(b). We propose S∗(b) as an approximation to S(b), and present a bound on the difference between them. Under certain circumstances, our finding is an improvement of a result of Alsmeyer, both in sharpness of the bound and in extension to more general sequences {Zj}. The methods employed are Tauberian in nature.