Total Variation Asymptotics for Sums of Independent Integer Random Variables

Let $W_n := \sum_{j=1}^n Z_j$ be a sum of independent integer-valued random variables. In this paper, we derive an asymptotic expansion for the probability $\mathbb{P}[W_n \in A]$ of an arbitrary subset $A \in \mathbb{Z}$. 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 $d_{\mathrm{TV}}(\mathscr{L}(Z_j), \mathscr{L}(Z_j + 1))$. The proofs are based on Stein's method for signed compound Poisson approximation. The Annals of Probability 2002, Vol. 30, No. 2, 509–545 TOTAL VARIATION ASYMPTOTICS FOR SUMS OF INDEPENDENT INTEGER RANDOM VARIABLES BY A. D. BARBOUR1 AND V. ČEKANAVIČIUS Universität Zürich and Vilnius University Let Wn := ∑nj=1 Zj be a sum of independent integer-valued random variables. In this paper, we derive an asymptotic expansion for the probability P[Wn ∈ A] of an arbitrary subset A ∈ 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 Zj : an appropriate number of moments and the total variation distance dTV(L(Zj ), L(Zj + 1)). The proofs are based on Stein’s method for signed compound Poisson approximation.