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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).

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[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)

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

Citations:

Zbl 0722.00031
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