Closed Newton-Cotes Quadrature Rules for Stieltjes Integrals and Numerical Convolution of Life Distributions

We propose a simple new algorithm for numerical conovolution of distributions having support on the nonnegative reals. The algorithm is based on a Simpson-like quadrature rule that deals directly with the Stieltjes integrals. It avoids having to assume the existence of a density for any of the distributions involved. 

Multiple convolutions are easily accommodated. We generalize the closed Newton-Cotes quadrature rules to Stieltjes integrals, and develop some of their properties. Straightforward generalization of the classical error bounds leads to bounds involving derivatives of the integrator function. We also provide error bounds in case the integrator function is only assumed to be of bounded variation over the interval of integration. 

We discuss the role of these error bounds in computations where the distributions convolved are known only in tabular form. Many applied probability computations can use this technique. Two important examples are reliability analysis of a cold standby redundant system, and in obtaining the distribution of the number of events in a renewal counting process.