From Gaussian to stable and back: Non-Pareto points in growing dimension

Abstract

We study the number of non-Pareto points in two high-dimensional models: a homogeneous Poisson point process and a fixed-size sample of independent uniform points in the unit cube. A point is non-Pareto if some other sample point is coordinatewise less than or equal to it. As the dimension grows on the logarithmic scale of the intensity or sample size, the asymptotic behavior exhibits a surprising sequence of transitions: from Gaussian to stable fluctuations, back to Gaussian, then to a Poisson limit at the sparsity threshold, followed by the eventual disappearance of non-Pareto points. We identify the critical thresholds, normalizations, and limiting laws in both models. Notably, the return to Gaussianity occurs while the stability index remains strictly below $2$ and is driven by a change in the dominant fluctuation mechanism. Moreover, in part of the returning Gaussian regime, the natural normalization is the square root of the expected count rather than its standard deviation. At the stable--Gaussian transition, the two mechanisms can coexist, giving rise to a sum of independent stable and Gaussian random variables.

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BibTeX RIS

BibTeX

@misc{bhattacharjee2026from,
  title = {{From Gaussian to stable and back: Non-Pareto points in growing dimension}},
  author = {Chinmoy Bhattacharjee and Andrii Ilienko and Bochen Jin and Ilya Molchanov},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01096},
  eprint = {2611.01096},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01096},
}

RIS

TY  - UNPB
TI  - From Gaussian to stable and back: Non-Pareto points in growing dimension
AU  - Chinmoy Bhattacharjee
AU  - Andrii Ilienko
AU  - Bochen Jin
AU  - Ilya Molchanov
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01096
UR  - https://arxiv.org/abs/2611.01096
AB  - We study the number of non-Pareto points in two high-dimensional models: a homogeneous Poisson point process and a fixed-size sample of independent uniform points in the unit cube. A point is non-Pareto if some other sample point is coordinatewise less than or equal to it. As the dimension grows on the logarithmic scale of the intensity or sample size, the asymptotic behavior exhibits a surprising sequence of transitions: from Gaussian to stable fluctuations, back to Gaussian, then to a Poisson limit at the sparsity threshold, followed by the eventual disappearance of non-Pareto points. We identify the critical thresholds, normalizations, and limiting laws in both models. Notably, the return to Gaussianity occurs while the stability index remains strictly below $2$ and is driven by a change in the dominant fluctuation mechanism. Moreover, in part of the returning Gaussian regime, the natural normalization is the square root of the expected count rather than its standard deviation. At the stable--Gaussian transition, the two mechanisms can coexist, giving rise to a sum of independent stable and Gaussian random variables.
ER  -

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