Approximate independence of typical marginals of high-dimensional probability measures
Abstract
For many probability measures on ℝn, and for k sufficiently smaller than n, most k-dimensional coordinate marginals are close to product measures. We prove this in two regimes, which notably involve no exchangeability assumptions as in prior theorems of de Finetti type. First, for log-concave measures, the quantitative bound depends on the measure only through the operator norm of its correlation matrix, and k may grow polynomially with n. A key ingredient of the proof is a new strengthening of the subadditivity of relative entropy for strongly log-concave measures, which as a byproduct yields sharp bounds in the strongly log-concave case. Second, for a broad class of measures satisfying just a variance inequality much weaker than Poincaré's, k must stay below the sharp threshold 2log n/log log n. Here the basis of ℝn is fixed a priori, which makes this phenomenon quite different from known Gaussian approximations for low-dimensional projections such as the central limit theorem for convex bodies, although there are remarkable parallels. Insofar as Gaussian approximations are driven by the thin shell property, our results are driven by variance inequalities for additive functions which have appeared in the literature on the Gibbs sampler.
Related
- How far are d-dimensional copulas with uniform (d−1)-marginals from (total) independence?
- SCMO: Stochastic Control for Optimization over Probability Measures on Infinite-Dimensional Spaces
- Degree-Free Spectral Independence for Log-Concave Holant Measures
- Ergodic measures of intermediate entropies for amenable group actions with an approximate product property
- Generalized Bratteli diagrams without probability tail invariant measures
BibTeX
@misc{arnese2026approximate,
title = {{Approximate independence of typical marginals of high-dimensional probability measures}},
author = {Manuel Arnese and Daniel Lacker},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07460},
eprint = {2611.07460},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07460},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Approximate independence of typical marginals of high-dimensional probability measures
AU - Manuel Arnese
AU - Daniel Lacker
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07460
UR - https://arxiv.org/abs/2611.07460
AB - For many probability measures on ℝ^n, and for k sufficiently smaller than n, most k-dimensional coordinate marginals are close to product measures. We prove this in two regimes, which notably involve no exchangeability assumptions as in prior theorems of de Finetti type. First, for log-concave measures, the quantitative bound depends on the measure only through the operator norm of its correlation matrix, and k may grow polynomially with n. A key ingredient of the proof is a new strengthening of the subadditivity of relative entropy for strongly log-concave measures, which as a byproduct yields sharp bounds in the strongly log-concave case. Second, for a broad class of measures satisfying just a variance inequality much weaker than Poincaré's, k must stay below the sharp threshold 2log n/log log n. Here the basis of ℝ^n is fixed a priori, which makes this phenomenon quite different from known Gaussian approximations for low-dimensional projections such as the central limit theorem for convex bodies, although there are remarkable parallels. Insofar as Gaussian approximations are driven by the thin shell property, our results are driven by variance inequalities for additive functions which have appeared in the literature on the Gibbs sampler.
ER -