Some results on shuffles of two-dimensional copulas

Using the one-to-one correspondence between two-dimensional copulas and special Markov kernels allows to study properties of T-shuffles of copulas, T being a general Lebesgue-measure-preserving transformation on [ 0 , 1 ] , in terms of the corresponding operation on Markov kernels. As one direct consequence of this fact the asymptotic behaviour of iterated T-shuffles S T n ( A ) of a copula A ∈ C can be characterized through mixing properties of T. In particular it is shown that S T n ( A ) ( ( 1 / n ) ∑ i = 1 n S T i ( A ) ) converges uniformly to the product copula Π for every copula A if and only if T is strongly mixing (ergodic). Moreover working with Markov kernels also allows, firstly, to give a short proof of the fact that the mass of the singular component of S T ( A ) cannot be bigger than the mass of the singular component of A, secondly, to introduce and study another operator U T : C → C fulfilling S T ○ U T ( A ) = A for all A ∈ C , and thirdly to express S T ( A ) and U T ( A ) as ⁎-product of A with the completely dependent copula CT induced by T.