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Lower cone distribution functions and set-valued quantiles form Galois connections. (English. Russian original) Zbl 1459.60027

Theory Probab. Appl. 65, No. 2, 179-190 (2020); translation from Teor. Veroyatn. Primen. 65, No. 2, 221-236 (2020).
Summary: It is shown that a recently introduced lower cone distribution function, together with the set-valued multivariate quantile, generates a Galois connection between a complete lattice of closed convex sets and the interval \([0,1]\). This generalizes the corresponding univariate result. It is also shown that an extension of the lower cone distribution function and the set-valued quantile characterize the capacity functional of a random set extension of the original multivariate variable along with its distribution.

MSC:

60E05 Probability distributions: general theory
62H05 Characterization and structure theory for multivariate probability distributions; copulas
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