Some properties of double shuffles of bivariate copulas and (extreme) copulas invariant with respect to Lüroth double shuffles

Abstract

Considering the well-known shuffling operation in x- and in y-direction yields so-called double shuffles of bivariate copulas. We study continuity properties of the double shuffle operator ST induced by pairs T=(T1×T2) of measure preserving transformations on ([0,1],B([0,1]),λ) on the family C of all bivariate copulas, analyze its interrelation with the star/Markov product, and show that for each left- and for each right-invertible copula A the set of all possible double shuffles of A is dense in C with respect to the uniform metric d∞. After deriving some general properties of the set ΩT of all ST-invariant copulas we focus on the situation where T1,T2 are strongly mixing and show that in this case the product copula Π is an extreme point of ΩT. Moreover, motivated by a recent paper by Horanská and Sarkoci (Fuzzy Sets and Systems 378, 2018) we then study double shuffles induced by pairs of so-called Lüroth maps and derive various additional properties of ΩT, including the surprising fact that ΩT contains uncountably many extreme points which (interpreted as doubly stochastic measures) are pairwise mutually singular with respect to each other and which allow for an explicit construction.