Approximations to generalized renewal measures

Abstract

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}, 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.