A Note on Bounds for the Supremum Metric for Discrete Random Variables

For positive or negative integer-valued random variables X and Y with finite second moments the inequality sup documentclass{article}pagestyle{empty}begin{document}$ mathop {sup }limits_n |Pr { X le n} - Pr { Y le n} |, le ,|EX - EY| + frac{1}{2}(EX(EX - 1) + (EY(Y - 1)) $end{document} is established by elementary manipulation, and shown to be tight. Use of generating functions and an inversion formula yields the larger bound with 1/2 replaced by 2/π.