On a class of extremal problems in statistics

Abstract

Let m denote the infimum of the Integral of a function q w r t all probability measures with given marginals. The determination of m is of interest for a series of stochastic problems. In the present paper we prove a duality theorem for the determination of m and give some examples for its application. We consider especially the problem of extremal variance of sums of random variables and prove a theorem for the existence of random variables with given marginal distributions, such that their sum has variance zero.