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

Alonso–Gutiérrez, David; Prochno, Joscha

doi:10.1016/j.jfa.2021.109291

Cited by 5 publications

(2 citation statements)

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“…Recently, Kabluchko and Prochno [31], using maximum entropy considerations in the framework of non-interacting particles that have their origin in statistical mechanics, derived an asymptotic version of a Schechtman-Zinn type representation for Orlicz spaces, relating the uniform distribution on Orlicz balls to Gibbs distributions with potential given by the Orlicz functions. This connection allowed to obtain a number of further results in the general setting of Orlicz balls, wich can be found, for instance, in [2,8,21,28,35].…”

Section: Introductionmentioning

confidence: 88%

“…The motivation of our paper is thus twofold, namely to develop a first probabilistic approach on the one hand and apply this to study some asymptotic properties of Lorentz spaces n q,p on the other. In view of the vast literature regarding a probabilistic take on n p or more generally n M spaces and their unit balls, this is a first step towards approaching numerous natural questions that arise, like central limit theorems, conditional limit theorems, or moderate and large deviations [3,4,21,22,30,32,36], asymptotic volume ratios and thin-shell measure concentration [2,31], asymptotic independence of coordinates [8] or asymptotic thin-shell results [35], just to mention a few.…”

Section: Introductionmentioning

confidence: 99%

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A probabilistic approach to Lorentz balls

Kabluchko¹,

Prochno²,

Sonnleitner³

2023

Preprint

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We develop a probabilistic approach to study the volumetric and geometric properties of unit balls B n q,1 of finite-dimensional Lorentz sequences spaces n q,1 . More precisely, we show that the empirical distribution of a random vector X (n) uniformly distributed on the volume normalized Lorentz ball in R n converges weakly to a compactly supported symmetric probability distribution with explicitly given density; as a consequence we obtain a weak Poincaré-Maxwell-Borel principle for any fixed number k ∈ N of coordinates of X (n) as n → ∞. Moreover, we prove a central limit theorem for the largest coordinate of X (n) , demonstrating a quite different behavior than in the case of the n q balls, where a Gumbel distribution appears in the limit. Last but not least, we prove a Schechtman-Schmuckenschläger type result for the asymptotic volume of intersections of volume normalized Lorentz and n p balls.

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