THE FIRST DIVISIBLE SUM

We consider the distribution of the first sum of a sequence of positiv e integer valued lid random variables which is divisible by d. This is known to converge, when divided by d, to a geometric distribution as d --> infinity. We obtain results on the rate of convergence using two contrasting approaches. In the first, Stein's method is adapted to ge ometric limit distributions. The second method is based on the theory of Banach algebras. Each method is shown to have its merits.