On the Maximum of a Measure of Deviation from Independence Between Discrete Random Variables

Abstract

The squared $n^k$-dimensional Euclidean distance $f_k$ between a given joint distribution of $k$ random variables with values in $1, \cdots, n$ and the joint distribution of independent variables with the same respective marginals has been suggested as a measure of dependence. The following facts are established for $M_k$, the maximum of $f_k$ over all joint distributions for fixed $k$: (1) $M_k$ is attained among the distributions with all $k$ variables equal to a variable $X$ that takes on just two values. (2) For $k \leq 6, M_k = 1/2 - (1/2)^k$ is attained when the distribution of $X$ is $\{1/2, 1/2\}$. (3) For $k \geq 7, M_k$ is not attained at $\{1/2, 1/2\}$ and strictly exceeds $1/2 - (1/2)^k$. (4) For $k \rightarrow \infty$, the distributions of $X$ where $M_k$ is attained approach $\{0, 1\}$, and $M_k \nearrow 1$.