Some results on shuffles of two-dimensional copulas

Abstract

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