Approximations to generalized renewal measures

Let {Z_j,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 as S(b)=∑_j=1^∞g(j)P{Z_j<b}, 0. Due to possible complexity of calculating the probabilities P{Z_j<b}, computation of S(b) is often intractable. Consider a sequence of positive numbers {m_j, j≥1} and define S^*(b)=∑_j=1^∞ g(j)I{m_j<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 {Z_j}. The methods employed are Tauberian in nature.