On diagonal sections of Markov products of copulas

Abstract

Given a bivariate copula $C$, the bivariate copula $Ψ(C)$ is defined as the Markov product of $C^T$ and $C$. In this paper we consider the diagonal section of $Ψ(C)$ under the additional assumption that its value at a single point is known. We determine the exact upper and lower bounds for the diagonal section of $Ψ(C)$, given that $Ψ(C)(a,a)=d \in [a^2,a]$, where $a \in [0,1]$ is a fixed point. Furthermore, we describe some additional properties of the transformation $Ψ$, such as its behavior under ordinal sums and the survival operation. We also study checkerboard copulas in the range of $Ψ$ and matrices associated to them.