Convergence of Differential Entropies -- II

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: $\sup_n \int f_n\, \Psi(|\log f_n|) < \infty$ for a single superlinear~$\Psi$, strictly weaker than the fixed-$\alpha$ condition of Godavarti and Hero (2004). We also disprove the Godavarti-Hero conjecture that $\alpha > 1$ could be replaced by $\alpha_n \downarrow 1$. 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.