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

Abstract

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 (finite) discrete random variables $\mathsf{X}$ and $\mathsf{Y}$ decreases at least quadratically in the Euclidean distance as $p_{\mathsf{X}}$ 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-Poor-Verdú [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.

Submitted to IEEE Transactions on Information Theory on Oct. 27th, 2022. Revised on March 7th, 2023. Accepted on April 30th 2023

References in corpus (1)

• A Tight Upper Bound for the Third-Order Asymptotics for Most Discrete Memoryless Channels

Cited by in corpus (1)

• The Mutual Information In The Vicinity of Capacity-Achieving Input Distributions