Total variation asymptotics for sums of independent integer random variables

Let W-n := Sigma(j-1)(n) (Zj) be a sum of independent integer-valued random variables. In this paper, we derive an asymptotic expansion for the probability P[W-n is an element of A] of an arbitrary subset A is an element of Z. Our approximation improves upon the classical expansions by including an explicit, uniform error estimate, involving only easily computable properties of the distributions of the Z(j): an appropriate number of moments and the total variation distance d(TV) (X (Zj), L(Z(j) + 1)). The proofs are based on Stein's method for signed compound Poisson approximation.