Convergence of Differential Entropies -- II

Mahesh Godavarti

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arXiv · 2602.13493

Convergence of Differential Entropies -- II

Mahesh Godavarti

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Abstract

We show that under convergence in measure of probability density functions, differential entropy converges whenever the entropy integrands $f_n |\log f_n|$ are uniformly integrable and tight -- a direct consequence of Vitali's convergence theorem. We give an entropy-weighted Orlicz condition: $\sup_n \int f_n\, Ψ(|\log f_n|) < \infty$ for a single superlinear~$Ψ$, strictly weaker than the fixed-$α$ condition of Godavarti and Hero (2004). We also disprove the Godavarti-Hero conjecture that $α> 1$ could be replaced by $α_n \downarrow 1$. We recover the sufficient conditions of Godavarti--Hero, Piera--Parada, and Ghourchian-Gohari-Amini as corollaries. On bounded domains, we prove that uniform integrability of the entropy integrands is both necessary and sufficient -- a complete characterization of entropy convergence.

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Mahesh Godavarti. 2026-02-18. Convergence of Differential Entropies -- II. https://arxiv.org/abs/2602.13493

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