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The quadratic decaying property of the information rate function states that given a fixed conditional distribution  p  Y|X , the mutual information between the (finite) discrete random variables X and Y decreases at least quadratically in the Euclidean distance as  p  X  moves away from the capacity-achieving input distributions. It is a 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 already implicitly used by Strassen [1] and later, more explicitly, by Polyanskiy&#x2013;Poor&#x2013;Verd&#x00FA; [2]. However, the proofs outlined in both works contain gaps that are nontrivial to close. This comment provides an alternative, complete proof of this property.

Comments on “Channel Coding Rate in the Finite Blocklength Regime”: On the Quadratic Decaying Property of the Information Rate Function