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Extreme points of sets of probability measures and $φ$-divergences — arXiv:2608.13358

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.13358
License (as archived): CC0-1.0
Archived: 2026-10-19T04:47:18.714Z
SHA-256 of the archived text: d355aa0caaf60fc0e421981492e4e8a67b210da90e86509990072c48d2ac74f0
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Extreme points of sets of probability measures and $φ$-divergences — arXiv:2608.13358 — arXiv (en), https://arxiv.org/abs/2608.13358 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_808714e697ef (Metraipolis archive dc_808714e697ef, sha256 d355aa0caaf60fc0e421981492e4e8a67b210da90e86509990072c48d2ac74f0)

BibTeX

@misc{dc808714e697ef,
  title = {Extreme points of sets of probability measures and $φ$-divergences — arXiv:2608.13358},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.13358},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_808714e697ef; sha256 d355aa0caaf60fc0e421981492e4e8a67b210da90e86509990072c48d2ac74f0},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.13358. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In this work, we prove several equivalent characterizations of the extreme points of convex sets of probability measures of the form $\mathcal{M}=\mathcal{P} \cap H$, where $\mathcal{P}$ denotes the set of all probability measures on an arbitrary measurable space $(\Omega,\mathcal{F})$ and $H$ is an affine set of signed measures on $(\Omega,\mathcal{F})$ with finite variation. We first give a precise measure-theoretic formulation of the heuristic that extreme measures have minimal support. We then connect this with the notion of minimality with respect to absolute continuity, and prove that points that dominate no other element of $\mathcal{M}$ are the only ones realizing the blow-up of a certain divergence map for any suitable $\phi$-divergence. Finally, considering a different class of $\phi$-divergences, we recover a characterization of the extreme points of $\mathcal{M}$ as strict local maximizers of $\phi$-divergences relative to any suitable dominating measures. We apply our result to recover and complement results from the literature in the context of finite spaces, sets of measures defined by integral constraints, multi-marginal couplings, and dominated sets of probability measures.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-14 in math.PR).
Categorie arXiv: Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2608.13358 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-19T04:27:45.058Z dalla pagina https://arxiv.org/catchup/math.PR/2026-08-14?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).