Thin-shell concentration for random vectors in Orlicz balls via moderate deviations and Gibbs measures

Alonso Gutiérrez, David (Universidad de Zaragoza) ; Prochno, Joscha

Resumen: In this paper, we study the asymptotic thin-shell width concentration for random vectors uniformly distributed in Orlicz balls. We provide both asymptotic upper and lower bounds on the probability of such a random vector $X_n$ being in a thin shell of radius $\sqrt{n}$ times the asymptotic value of $n^{-1/2}\left(\E\left[\Vert X_n\Vert_2^2\right]\right)^{1/2}$ (as $n\to\infty$), showing that in certain ranges our estimates are optimal. In particular, our estimates significantly improve upon the currently best known general Lee-Vempala bound when the deviation parameter $t=t_n$ goes down to zero as the dimension $n$ of the ambient space increases. We shall also determine in this work the precise asymptotic value of the isotropic constant for Orlicz balls. Our approach is based on moderate deviation principles and a connection between the uniform distribution on Orlicz balls and Gibbs measures at certain critical inverse temperatures with potentials given by Orlicz functions, an idea recently presented by Kabluchko and Prochno in [The maximum entropy principle and volumetric properties of Orlicz balls, J. Math. Anal. Appl. {\bf 495}(1) 2021, 1--19].

Idioma: Inglés

DOI: 10.1016/j.jfa.2021.109291

Año: 2022

Publicado en: JOURNAL OF FUNCTIONAL ANALYSIS 282, 1 (2022), 109291 [35 pp.]
ISSN: 0022-1236

Factor impacto JCR: 1.7 (2022)

Categ. JCR: MATHEMATICS rank: 51 / 329 = 0.155 (2022) - Q1 - T1

Factor impacto CITESCORE: 2.9 - Mathematics (Q2)

Factor impacto SCIMAGO: 1.959 - Analysis (Q1)

Financiación: info:eu-repo/grantAgreement/ES/DGA/E48-20R
Financiación: info:eu-repo/grantAgreement/ES/MICINN PID2019-105979GB-I00

Tipo y forma: Artículo (PostPrint)

Área (Departamento): Área Análisis Matemático (Dpto. Matemáticas)