Abstract:Majorization is a stochastic ordering relation that compares the relative diversity of probability distributions with numerous applications in econometrics, spectral theory, and ecology. It is well-known that the majorization partial order forms a complete lattice on the set of ordered probability distributions. In this work, we study the properties of Rényi entropy on the majorization lattice. We establish a fundamental relation between the comonotone coupling and the independent coupling associated with a collection of marginal distributions. Consequently, we show that, for every order $\alpha \in [0,\infty]$, the Rényi entropy is subadditive on the majorization lattice. We further characterize the supermodular regime, showing that Rényi entropy is supermodular on the majorization lattice for $\alpha \in \{0\} \,\cup \, [1,\infty]$. For the Tsallis entropy, we show that it also satisfies subadditivity on the majorization lattice, for every order $\alpha \in [0,\infty)$. Finally, we show that, unlike the Rényi entropy, the Tsallis entropy is supermodular on the majorization lattice for every $\alpha \in [0,\infty)$.
| Comments: | 20 pages, 2 figures |
| Subjects: | Information Theory (cs.IT); Combinatorics (math.CO); Probability (math.PR) |
| Cite as: | arXiv:2605.09655 [cs.IT] |
| (or arXiv:2605.09655v2 [cs.IT] for this version) | |
| https://doi.org/10.48550/arXiv.2605.09655 arXiv-issued DOI via DataCite |
From: Anuj Kumar Yadav [view email]
[v1]
Sun, 10 May 2026 16:58:44 UTC (178 KB)
[v2]
Fri, 22 May 2026 14:38:40 UTC (190 KB)
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