On the Quadratic Decaying Property of the Information Rate Function

The quadratic decaying property of the information rate function states that given a fixed conditional distribution $p_{\mathsf{Y}|\mathsf{X}}$, the mutual information between the random variables $\mathsf{X}$ and $\mathsf{Y}$ decreases at least quadratically in the distance as $p_\mathsf{X}$ moves away from the capacity-achieving input distributions. It is a fundamental property of the information rate function that is particularly useful in the study of higher order asymptotics and finite blocklength information theory, where it was first used by Strassen [1] and later, more explicitly, Polyanskiy-Poor-Verd\'u [2], [3]. Recently, while applying this result in our work, we were not able to close apparent gaps in both of these proofs. This motivates us to provide an alternative proof in this note.