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

Let X 1, X 2,..., X n and Y 1, Y 2,..., Y n 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 X i ’s or Y i ’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.