Error analysis of some integration procedures for renewal equation and convolution integrals

When computing convolution integrals such as needed in solving renewal-type integral equations, direct integration of the Riemann-Stieltjes integral can be used. This method has been shown to be simple and accurate with good convergence property. In this paper bounds on the error terms are derived for some direct Riemann-Stieltjes integration methods. The bounds are rough but they are given under very weak assumptions. The bounds are further improved when certain derivability conditions are satisfied. It is shown that, under some continuity assumptions, the convergence is of order 2 as for the case of traditional Riemann integral. This provides the theoretical foundation to the direct Riemann-Stieltjes integration which can be used to approximate Riemann-Stieltjes integrals and in solving renewal-type equations.