A probabilistic approach to Lorentz balls
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916401656102912 |
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| author | Kabluchko, Zakhar Prochno, Joscha Sonnleitner, Mathias |
| author_facet | Kabluchko, Zakhar Prochno, Joscha Sonnleitner, Mathias |
| contents | We develop a probabilistic approach to study the volumetric and geometric properties of unit balls $\mathbb B_{q,1}^n$ of finite-dimensional Lorentz sequences spaces $\ell_{q,1}^n$. More precisely, we show that the empirical distribution of a random vector $X^{(n)}$ uniformly distributed on the volume normalized Lorentz ball in $\mathbb R^n$ 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 $k\in\mathbb N$ of coordinates of $X^{(n)}$ as $n\to\infty$. 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 $\ell_q^n$ 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 $\ell^n_p$ balls. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_04728 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A probabilistic approach to Lorentz balls Kabluchko, Zakhar Prochno, Joscha Sonnleitner, Mathias Functional Analysis Probability Primary 46B09, 52A23, 60F05, Secondary 46B06, 46B20, 46B45, 94A17 We develop a probabilistic approach to study the volumetric and geometric properties of unit balls $\mathbb B_{q,1}^n$ of finite-dimensional Lorentz sequences spaces $\ell_{q,1}^n$. More precisely, we show that the empirical distribution of a random vector $X^{(n)}$ uniformly distributed on the volume normalized Lorentz ball in $\mathbb R^n$ 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 $k\in\mathbb N$ of coordinates of $X^{(n)}$ as $n\to\infty$. 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 $\ell_q^n$ 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 $\ell^n_p$ balls. |
| title | A probabilistic approach to Lorentz balls |
| topic | Functional Analysis Probability Primary 46B09, 52A23, 60F05, Secondary 46B06, 46B20, 46B45, 94A17 |
| url | https://arxiv.org/abs/2303.04728 |