Limit Theorems for the Volume of Random Projections and Sections of $\ell_p^N$-balls
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arXiv
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2024
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| _version_ | 1866917875602685952 |
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| author | Prochno, Joscha Thaele, Christoph Tuchel, Philipp |
| author_facet | Prochno, Joscha Thaele, Christoph Tuchel, Philipp |
| contents | Let $\mathbb{B}_p^N$ be the $N$-dimensional unit ball corresponding to the $\ell_p$-norm. For each $N\in\mathbb N$ we sample a uniform random subspace $E_N$ of fixed dimension $m\in\mathbb{N}$ and consider the volume of $\mathbb{B}_p^N$ projected onto $E_N$ or intersected with $E_N$. We also consider geometric quantities other than the volume such as the intrinsic volumes or the dual volumes. In this setting we prove central limit theorems, moderate deviation principles, and large deviation principles as $N\to\infty$. Our results provide a complete asymptotic picture. In particular, they generalize and complement a result of Paouris, Pivovarov, and Zinn [A central limit theorem for projections of the cube, Probab. Theory Related Fields. 159 (2014), 701-719] and another result of Adamczak, Pivovarov, and Simanjuntak [Limit theorems for the volumes of small codimensional random sections of $\ell_p^n$-balls, Ann. Probab. 52 (2024), 93-126]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_16054 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Limit Theorems for the Volume of Random Projections and Sections of $\ell_p^N$-balls Prochno, Joscha Thaele, Christoph Tuchel, Philipp Probability Metric Geometry Primary 52A23, 60F05, 60F10, Secondary 46B09, 52A22, 60D05 Let $\mathbb{B}_p^N$ be the $N$-dimensional unit ball corresponding to the $\ell_p$-norm. For each $N\in\mathbb N$ we sample a uniform random subspace $E_N$ of fixed dimension $m\in\mathbb{N}$ and consider the volume of $\mathbb{B}_p^N$ projected onto $E_N$ or intersected with $E_N$. We also consider geometric quantities other than the volume such as the intrinsic volumes or the dual volumes. In this setting we prove central limit theorems, moderate deviation principles, and large deviation principles as $N\to\infty$. Our results provide a complete asymptotic picture. In particular, they generalize and complement a result of Paouris, Pivovarov, and Zinn [A central limit theorem for projections of the cube, Probab. Theory Related Fields. 159 (2014), 701-719] and another result of Adamczak, Pivovarov, and Simanjuntak [Limit theorems for the volumes of small codimensional random sections of $\ell_p^n$-balls, Ann. Probab. 52 (2024), 93-126]. |
| title | Limit Theorems for the Volume of Random Projections and Sections of $\ell_p^N$-balls |
| topic | Probability Metric Geometry Primary 52A23, 60F05, 60F10, Secondary 46B09, 52A22, 60D05 |
| url | https://arxiv.org/abs/2412.16054 |