Convergence of Differential Entropies -- II

We show that under convergence in measure of probability density functions, differential entropy converges whenever the entropy integrands $f_n\mathrm{\mid }\log f_n\mathrm{\mid }$ are uniformly integrable and tight – a direct consequence of Vitali’s convergence theorem. We give an entropy-weighted Orlicz condition: ${\sup }_n\int f_n\mathrm{\Psi }(\mathrm{\mid }\log f_n\mathrm{\mid })<\mathrm{\infty }$ for a single superlinear~$\mathrm{\Psi }$, 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.