An Optimal Upper Bound on the Tail Probability for Sums of Random Variables

Abstract

Let $s$ be any given real number. An explicit construction is provided of random variables (r.v.'s) $X$ and $Y$ such that $\\sup{P}(X+Y\\ge s)$ is attained, where the $\\sup$ is taken over all r.v.'s $X$ and $Y$ with given distributions.