Sudakov minoration for unconditional log-concave vectors with negatively associated magnitudes
Abstract
We prove the Sudakov minoration principle, with a universal constant, for unconditional log-concave random vectors whose coordinate magnitudes are negatively associated. The main step is a deterministic selection theorem: pairwise witnesses in convex downwardclosed sets yield, on an exponentially large subfamily, one fixed witness per label separating it from every other retained label. The selection combines a prefix counting inequality with convex separation in the space of all label-coordinate arrays. Negative association then controls the expected number of active joint-tail witnesses, while a Bernoulli argument on complementary coordinate sets retains the random signs. The proof uses joint upper-orthant probabilities and applies negative association only under the original law. Consequences include minoration for products of Orlicz-based log-concave measures and a covering estimate in the moment metric, expressed geometrically through polar Lp centroid bodies.
Related
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- Revisiting entropy and entropy-related magnitudes of a symbolic sequence and their estimation
BibTeX
@misc{bednorz2026sudakov,
title = {{Sudakov minoration for unconditional log-concave vectors with negatively associated magnitudes}},
author = {Witold Bednorz and Rafal Martynek and Rafal Meller},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30525},
eprint = {2609.30525},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30525},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sudakov minoration for unconditional log-concave vectors with negatively associated magnitudes
AU - Witold Bednorz
AU - Rafal Martynek
AU - Rafal Meller
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30525
UR - https://arxiv.org/abs/2609.30525
AB - We prove the Sudakov minoration principle, with a universal constant, for unconditional log-concave random vectors whose coordinate magnitudes are negatively associated. The main step is a deterministic selection theorem: pairwise witnesses in convex downwardclosed sets yield, on an exponentially large subfamily, one fixed witness per label separating it from every other retained label. The selection combines a prefix counting inequality with convex separation in the space of all label-coordinate arrays. Negative association then controls the expected number of active joint-tail witnesses, while a Bernoulli argument on complementary coordinate sets retains the random signs. The proof uses joint upper-orthant probabilities and applies negative association only under the original law. Consequences include minoration for products of Orlicz-based log-concave measures and a covering estimate in the moment metric, expressed geometrically through polar Lp centroid bodies.
ER -