Rüschendorf, Ludger Fréchet-bounds and their applications. (English) Zbl 0744.60005 Advances in probability distributions with given marginals. Beyond the copulas, Lect. Symp., Rome/Italy, Math. Appl. 67, 151-187 (1991). Show indexed articles as search result. [For the entire collection see Zbl 0722.00031.]This paper gives a review of Fréchet bounds, their applications, and connections to various topics in pure and applied mathematics. The bibliography contains 112 positions. The topic stems from classical problems of existence, uniqueness, and constructibility of probability measures on a product space, given a collection of marginal probabilities, subject to a suitable consistency assumption. There are defined several versions of this problem, beginning with the most common one, when one-dimensional marginals are given, and ending with partially ordered families of probability spaces.A quantitative side of the problem is represented by use of minimal metrics (surveyed in the paper) and, related to them, Fréchet bounds. The latter bounds involve functionals of the form \(M(\phi)=\sup(\hbox{or }\inf)\{\int\phi dP: \hbox{marginal }P\}\), where \(\phi\) are measurable real functions. The main issue here is the possibility of employing the duality theory, either by means of basic techniques in linear programming, or, in a more abstract form, Hahn-Banach theorem. A sample of typical applications is represented by Strassen’s stochastic orders, optimization problems, dependence structure of multivariate distributions, economics models, statistical inference and estimation, real analysis, etc. Reviewer: J.Szulga (Auburn) Cited in 26 Documents MSC: 60A10 Probabilistic measure theory 28A35 Measures and integrals in product spaces 60C05 Combinatorial probability 60B10 Convergence of probability measures 60E99 Distribution theory 62B99 Sufficiency and information 62H05 Characterization and structure theory for multivariate probability distributions; copulas 26D07 Inequalities involving other types of functions 26B25 Convexity of real functions of several variables, generalizations 46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators 46A20 Duality theory for topological vector spaces Keywords:Fréchet bounds; marginal distribution; marginal sufficiency; minimal metrics; probability metrics; product probability measure; Young inequality; Oppenheim inequality; stochastic order; Hahn-Banach theorem; dual theorems; duality; Strassen representation; Kantorovich problem; coupling; Huzurbazar conjecture; c-convex functions; probability measures on a product space; stochastic orders; dependence structure of multivariate distributions Citations:Zbl 0722.00031 PDFBibTeX XMLCite \textit{L. Rüschendorf}, in: Advances in probability distributions with given marginals. Beyond the copulas, Lect. Symp., Rome/Italy, Math. Appl. 67, . 151--187 (1991; Zbl 0744.60005)