Mutual Dependence of Random Variables and Maximum Discretized Entropy

Abstract

In connection with a random vector $(X, Y)$ in the unit square $Q$ and a couple $(m, n)$ of positive integers, we consider all discretizations of the continuous probability distribution of $(X, Y)$ that are obtained by an $m \times n$ cartesian decomposition of $Q$. We prove that $Y$ is a (continuous and invertible) function of $X$ if and only if for each $m, n$ the maximum entropy of the finite distributions equals $\log(m + n - 1)$