Limit Theorems Under Several Linear Constraints
Fabrice Gamboa, Martin Venker
TL;DR
The paper analyzes high-dimensional random vectors uniformly distributed on polytopes $K_n=\{x\ge0:Ax=b\}$ and shows that suitable linear projections are asymptotically Gaussian as $n\to\infty$, with variance given by $\sigma^2=\|P_{\ker(\hat{A})}\hat{\lambda}\|^2$. The authors develop a novel complex de Finetti-type representation, expressing $P_n$ as a mixture of product-like measures and derive a Bartlett-type formula for the characteristic function of the statistic $S_n$, enabling a central limit theorem under mild degeneracy control. A maximum-entropy principle identifies the probabilistic barycenter via $w=A^t\Lambda_0$, ensuring a convenient exponential-marginal approximation and centering of the constraint expectations. The work also proves a genericity result for the key assumption $\|\hat{A}\|_{\max}=o(1)$ and provides practical computational tools, including a simple Python routine, to compute the entropy center. Overall, the results sharpen understanding of constrained high-dimensional distributions, with implications for asymptotic geometry and probabilistic modeling under linear constraints.
Abstract
We study vectors chosen at random from a compact convex polytope in $\mathbb{R}^n$ given by a finite number of linear constraints. We determine which projections of these random vectors are asymptotically normal as $n\to\infty$. Marginal distributions are also studied, showing that in the large $n$ limit random variables under linear constraints become i.i.d. exponential under a rescaling. Our novel approach is based on a complex de Finetti theorem revealing an underlying independence structure as well as on entropy arguments.