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.

翻译：信息速率函数的二次衰减属性表示, 根据固定的有条件分配 $p ⁇ mathsf{Y ⁇ mathsf{X ⁇ $, 随机变量 $\mathsf{X} $ 和 $\mathsf{Y} 之间的相互信息在距离上至少递减四进制, 因为 $p ⁇ mathsf{X} 移动了实现输入分布的能力。 这是信息速率函数的一个基本属性, 在研究更高排序的无序和有限区段信息理论中特别有用, Strassen首先使用这个属性[1],后来更明确地说, Polyanskiy- Poor-Verd\'u [2], [3]。 最近, 在应用这个结果时, 我们无法弥补这两个证据的明显差距。 这促使我们在本注释中提供替代证据。