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PreprintarXiv licencearXiv

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

  • Weitao Huang
  • Renzhen Wang
  • Yichen Wu
  • 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 W1, both in expectation and almost surely along all sample sizes. The constant is c*=κ1/(2√π)∈(0,1/4], where κ1 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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Cite

BibTeXRIS
BibTeX
@misc{huang2026exact,
  title = {{Exact Endpoint Asymptotics and Almost-Sure Limits for Uniform Empirical Measures in Two Dimensions}},
  author = {Weitao Huang and Renzhen Wang and Yichen Wu and Deyu Meng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21070},
  eprint = {2610.21070},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21070},
}
RIS
TY  - UNPB
TI  - Exact Endpoint Asymptotics and Almost-Sure Limits for Uniform Empirical Measures in Two Dimensions
AU  - Weitao Huang
AU  - Renzhen Wang
AU  - Yichen Wu
AU  - Deyu Meng
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.21070
UR  - https://arxiv.org/abs/2610.21070
AB  - 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√π)∈(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.
ER  -