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arXiv · 2610.21070

Exact Endpoint Asymptotics and Almost-Sure Limits for Uniform Empirical Measures in Two Dimensions

Weitao Huang·Yichen Wu·Renzhen Wang·Deyu Meng
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Abstract

For the empirical measure of the first $n$ points of one infinite sequence of independent uniform samples in the unit square, we identify the leading constant in the bounded-Lipschitz distance to area measure, equivalently in $W_1$, both in expectation and almost surely along all sample sizes. The constant is $c_*=κ_1/(2\sqrtπ)\in(0,1/4]$, where $κ_1$ is the linear-growth Gaussian variational constant of Armegioiu--Goldman--Grotto--Trevisan. This answers the two-dimensional bounded-Lipschitz convergence question posed by Yukich in 1989. We give the complete endpoint argument, square--torus comparison, empirical reduction, and full-sequence probability argument.

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Weitao Huang, Yichen Wu, Renzhen Wang, Deyu Meng. 2026-10-15. Exact Endpoint Asymptotics and Almost-Sure Limits for Uniform Empirical Measures in Two Dimensions. https://arxiv.org/abs/2610.21070

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