Estimation and Approximation of Densities of IID sums via Importance Sampling

Sums of random variables appear frequently in several areas of the pure and applied sciences. When the variables are independent the sum density is the convolution of individual density functions. Convolution is almost always computationally intensive. We examine here the point estimation of i.i.d. sum densities and introduce the idea of an importance sampling convolver. This motivates an approximate analytical representation of the sum density that is easily computed. The representation involves a single convolution and is applicable to individual densities whose moment generating functions exist. Convergence to normality of the asymptotic form of the approximate density is established. The corresponding distribution approximations in the finite and asymptotic case are also given. One well-known application of practical value is considered in detail to demonstrate use of the approximation and establish its closeness to optimized simulation results. The key finding in this paper is that importance sampling can, in certain situations, lead to approximate formulae.