The first divisible sum

Abstract

We consider the distribution of the first sum of a sequence of positive integer valued iid random variables which is divisible byd. This is known to converge, when divided byd, to a geometric distribution asd→∞. We obtain results on the rate of convergence using two contrasting approaches. In the first, Stein's method is adapted to geometric limit distributions. The second method is based on the theory of Banach algebras. Each method is shown to have its merits.