Some Consequences of the Markov Kernel Perspective of Copulas

Abstract

The objective of this paper is twofold: After recalling the one-to-one correspondence between two-dimensional copulas and Markov kernels having the Lebesgue measure \(\lambda \) on \([0,1]\) as fixed point, we first give a quick survey over some consequences of this interrelation. In particular, we sketch how Markov kernels can be used for the construction of strong metrics that strictly distinguish extreme kinds of statistical dependence, and show how the translation of various well-known copula-related concepts to the Markov kernel setting opens the door to some surprising mathematical aspects of copulas. Secondly, we concentrate on the fact that iterates of the star product of a copula \(A\) with itself are Cesaro convergent to an idempotent copula \(\hat{A}\) with respect to any of the strong metrics mentioned before and prove that \(\hat{A}\) must have a very simple form if the Markov operator \(T_A\) associated with \(A\) is quasi-constrictive in the sense of Lasota.