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

Abstract

Recently, in answer to a question of Kolmogorov, G.D. Makarov obtained best-possible bounds for the distribution function of the sum X+Y of two random variables, X and Y, whose individual distribution functions, FX and FY, are fixed. We show that these bounds follow directly from an inequality which has been known for some time. The techniques we employ, which are based on copulas and their properties, 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. © 1987 Springer-Verlag.