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][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 STn(A)STn(A) of a copula A∈CA∈C can be characterized through mixing properties of T  . In particular it is shown that STn(A)STn(A) ((1/n)∑i=1nSTi(A)) converges uniformly to the product copula ΠΠ 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 ST(A)ST(A) cannot be bigger than the mass of the singular component of A  , secondly, to introduce and study another operator UT:C→CUT:C→C fulfilling ST○UT(A)=AST○UT(A)=A for all A∈CA∈C, and thirdly to express ST(A)ST(A) and UT(A)UT(A) as ⁎-product of A with the completely dependent copula CT induced by T.