On coupling of probability distributions and estimating the divergence through variation

Abstract

Let X be a discrete random variable with a given probability distribution. For any α, 0 ≤ α ≤ 1, we obtain precise values for both the maximum and minimum variational distance between X and another random variable Y under which an α-coupling of these random variables is possible. We also give the maximum and minimum values for couplings of X and Y provided that the variational distance between these random variables is fixed. As a consequence, we obtain a new lower bound on the divergence through variational distance.