Convergence of Differential Entropies -- II
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908838979960832 |
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| author | Godavarti, Mahesh |
| author_facet | Godavarti, Mahesh |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13493 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convergence of Differential Entropies -- II Godavarti, Mahesh Information Theory Probability 94A17, 28A20, 60B10 E.4; G.3 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. |
| title | Convergence of Differential Entropies -- II |
| topic | Information Theory Probability 94A17, 28A20, 60B10 E.4; G.3 |
| url | https://arxiv.org/abs/2602.13493 |