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

TL;DR

This paper provides a complete proof of the quadratic decaying property of the information rate function, which is crucial for understanding finite blocklength regimes in information theory.

Contribution

It offers a rigorous, alternative proof of a key property used in finite blocklength information theory, addressing gaps in previous proofs.

Findings

Confirmed the quadratic decay of the information rate function

Clarified the mathematical foundations of the property

Enhanced the theoretical understanding of finite blocklength regimes

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-Verd\'u [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.