A Central Limit Theorem For Random Variables Under A Constraint
Abstract
We consider vectors chosen randomly from high-dimensional hypersurfaces of the form {x∈ℝn: ∑j=1n Vj(xj)=an}. These in general asymmetric surfaces generalize classical spheres and include unbounded and disconnected surfaces. We show that under mild assumptions as n→∞, any weighted sum of the random coordinates is asymptotically Gaussian, provided that no weight dominates. An explicit reduction in the variance compared to the unconstrained case is observed. We also study a model with penalization of deviations from the constraint, interpolating between the unconstrained and the constrained case. Our approach involves an explicit representation of the characteristic function of the constrained random variables in terms of independent random variables.
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BibTeX
@misc{cromwell2026central,
title = {{A Central Limit Theorem For Random Variables Under A Constraint}},
author = {Jack Cromwell and Martin Venker},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08446},
eprint = {2611.08446},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08446},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - A Central Limit Theorem For Random Variables Under A Constraint
AU - Jack Cromwell
AU - Martin Venker
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08446
UR - https://arxiv.org/abs/2611.08446
AB - We consider vectors chosen randomly from high-dimensional hypersurfaces of the form {x∈ℝ^n: ∑_j=1^n V_j(x_j)=an}. These in general asymmetric surfaces generalize classical spheres and include unbounded and disconnected surfaces. We show that under mild assumptions as n→∞, any weighted sum of the random coordinates is asymptotically Gaussian, provided that no weight dominates. An explicit reduction in the variance compared to the unconstrained case is observed. We also study a model with penalization of deviations from the constraint, interpolating between the unconstrained and the constrained case. Our approach involves an explicit representation of the characteristic function of the constrained random variables in terms of independent random variables.
ER -