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Shuffles of Min. (English) Zbl 0768.60017
Summary: Copulas are functions which join the margins to produce a joint distribution function. A special class of copulas called shuffles of Min is shown to be dense in the collection of all copulas. Each shuffle of Min is interpreted probabilistically. Using the above-mentioned results, it is proved that the joint distribution of any two continuously distributed random variables $$X$$ and $$Y$$ can be approximated uniformly, arbitrarily closely by the joint distribution of another pair $$X^*$$ and $$Y^*$$ each of which is almost surely an invertible function of the other such that $$X$$ and $$X^*$$ are identically distributed as are $$Y$$ and $$Y^*$$. The preceding results shed light on A. Rényi’s axioms for a measure of dependence and a modification of those axioms as given by B. Schweizer and E. F. Wolff [Ann. Stat. 9, 879-885 (1981; Zbl 0468.62012)].

##### MSC:
 60E05 Probability distributions: general theory 60B10 Convergence of probability measures 62E10 Characterization and structure theory of statistical distributions 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62H20 Measures of association (correlation, canonical correlation, etc.)
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