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arXiv · 2611.01096

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

Chinmoy Bhattacharjee·Andrii Ilienko·Bochen Jin·Ilya Molchanov
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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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Chinmoy Bhattacharjee, Andrii Ilienko, Bochen Jin, Ilya Molchanov. 2026-10-31. From Gaussian to stable and back: Non-Pareto points in growing dimension. https://arxiv.org/abs/2611.01096

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