A New Method for Bounding the Distance Between Sums of Independent Integer-Valued Random Variables

Abstract

Let X1, X2,..., Xn and Y1, Y2,..., Yn be two sequences of independent random variables which take values in ℤ and have finite second moments. Using a new probabilistic method, upper bounds for the Kolmogorov and total variation distances between the distributions of the sums \(\sum_{i=1}^{n}X_{i}\) and \(\sum_{i=1}^{n}Y_{i}\) are proposed. These bounds adopt a simple closed form when the distributions of the coordinates are compared with respect to the convex order. Moreover, they include a factor which depends on the smoothness of the distribution of the sum of the Xi’s or Yi’s, in that way leading to sharp approximation error estimates, under appropriate conditions for the distribution parameters. Finally, specific examples, concerning approximation bounds for various discrete distributions, are presented for illustration.