Shuffles of Min. (English) Zbl 0768.60017

Mikusinski, P.; Sherwood, H.; Taylor, M. D.

Stochastica 13, No. 1, 61-74 (1992).

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.)

Keywords:

doubly stochastic measure; functional dependence; stochastic independence; copulas; measure of dependence