Metrika, Volume 30, 1983, page 55-61. Solution of a Statistical Optimization Problem by Rearrangement Methods By L. Riischendorf, Freiburg I ) Summary: Inequalities for the rearrangement of functions are applied to obtain a solution of a statistical optimization problem. This optimization problem arises in situations where one wants to describe the influence of stochastic dependence on a statistical problem. 1. Introduction LetP1 ..... Pn be n elements of M 1 (R 1, B 1) - the set of all probability measures on (R 1 , 81) - and define M (P1 ..... Pn) to be the set of all probability distributions on (R n, B n) with Pi as i-th marginal distribution, 1 ~< i ~<n. For measurable functions : R n ~ R 1 define m :=inf{fgdP:PEM(P1,... ,Pn)) (1) assuming the integrals in (1) to exist. M (Pt ..... Pn ) is convex, tight and closed and, therefore, by Prohorov's theorem compact w.r.t, weak-topology. To prove tightness, let K i E R 1, 1 <. i <. n, be compact sets with P. (K i) >t 1 -- e/n, I <<. i <<. n, then K = K1 • 9 9 9 X K n is compact and using n Fr6chet's lower bounds P (K) t> Z P; (K;) -- (n - 1)/> 1 -- e for all i= 1 " " P E M (P1, 9 9 9 Pn ). Therefore, the set on the right hand side of (1) is an interval and there exists a solution P* of (1) if ~ois bounded and continuous. (For bounded measu- rable r there exists in general only a solution in the set of normed additive set func- tions with marginals Pi' 1 <~ i <~ n.) Problem (1) arises in situations where one wants to describe the influence of dependence on a statistical problem. Some typical examples are the following ones: 1) Dr. L. Riischendor.f,, Institut ftit Mathematische Stochastik der Albert-Ludwigs-Universit~it, Hebelstr. 27, D-7800 Freiburg im Breisgau. 0026-1335/83/010055-6152.50 9 1983 Physica- Verlag, Vienna.