Limit Theorems Under Several Linear Constraints

We study $n$ real-valued random variables subject to several linear constraints.

Our main result is a weighted Central Limit Theorem, determining which linear combinations of these random variables 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.