Best-possible bounds for the distribution of a sum -- a problem of Kolmogorov (Q1071367)

Recently, in answer to a question of Kolmogorov, \textit{G. D. Makarov} [Theory Probab. Appl. 26, 803--806 (1981); translation from Teor. Veroyatn. Primen. 26, 815--817 (1981; Zbl 0474.60011)] obtained best- possible bounds for the distribution function of the sum \(X+Y\) of two random variables, \(X\) and \(Y\), whose individual distribution functions, \(F_ X\) and \(F_ Y\), are fixed. We show that these bounds follow directly from an inequality which has been known for some time. The techniques we employ yield an insightful proof of the fact that these bounds are best-possible, settle the question of equality, and are computationally manageable. Furthermore, they extend to binary operations other than addition and to higher dimensions.