Sharp concentration phenomena in high-dimensional Orlicz balls

Lorenz Frühwirth; Joscha Prochno

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arXiv · 2407.15579

Sharp concentration phenomena in high-dimensional Orlicz balls

Lorenz Frühwirth·Joscha Prochno

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Abstract

In this article, we present a precise deviation formula for the intersection of two Orlicz balls generated by Orlicz functions $V$ and $W$. Additionally, we establish a (quantitative) central limit theorem in the critical case and a strong law of large numbers for the "$W$-norm" of the uniform distribution on $\mathbb{B}^{(n,V)}$. Our techniques also enable us to derive a precise formula for the thin-shell concentration of uniformly distributed random vectors in high-dimensional Orlicz balls. In our approach we establish an Edgeworth-expansion using methods from harmonic analysis together with an exponential change of measure argument.

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Lorenz Frühwirth, Joscha Prochno. 2024-07-22. Sharp concentration phenomena in high-dimensional Orlicz balls. https://arxiv.org/abs/2407.15579

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