Large and Moderate Deviations for Entries of Orthogonal Matrices and the Stiefel Manifold
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| Formato: | Preprint |
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2025
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| _version_ | 1866918150493175808 |
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| author | Tuchel, Philipp |
| author_facet | Tuchel, Philipp |
| contents | Let $A_N$ be distributed according to the Haar probability measure on the orthogonal group $\mathscr{O}(N)$ for each $N\in\mathbb{N}$. It is well-known that the upper left $m_N\times k_N$ block of $\sqrt{N}A_N$ with $m_Nk_N = o(N)$ converges in total variation distance to a matrix of same size consisting of i.i.d. standard normal entries as $N\to\infty$. In this work, we characterize this convergence on the scale of large deviations. More precisely, we show that under the same condition $m_Nk_N = o(N)$ the empirical measure of entries of this block satisfies a large deviation principle with speed $m_Nk_N$ and rate function given by the relative entropy with respect to the standard normal distribution. Further, we complement the large deviation principle (LDP) obtained by Kabluchko and Prochno in [Large deviations for random matrices in the orthogonal group and Stiefel manifold with applications to random projections of product distributions, Annales de l'Institut Henri Poincaré. 60 (2024), 990 -- 1024] for the whole block $A_N$ with a moderate deviation principle (MDP). Concretely, we show an MDP for the sequence of matrices $β_N A_N$ in the product topology, where $β_N\to\infty$ is a sequence of real numbers such that $β_N = o(\sqrt{N})$. Here, in contrast to the LDP, the Gaussian behavior of the entries is reflected in the rate function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_24538 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Large and Moderate Deviations for Entries of Orthogonal Matrices and the Stiefel Manifold Tuchel, Philipp Probability Primary 60B20, 60F10, Secondary 15B52 Let $A_N$ be distributed according to the Haar probability measure on the orthogonal group $\mathscr{O}(N)$ for each $N\in\mathbb{N}$. It is well-known that the upper left $m_N\times k_N$ block of $\sqrt{N}A_N$ with $m_Nk_N = o(N)$ converges in total variation distance to a matrix of same size consisting of i.i.d. standard normal entries as $N\to\infty$. In this work, we characterize this convergence on the scale of large deviations. More precisely, we show that under the same condition $m_Nk_N = o(N)$ the empirical measure of entries of this block satisfies a large deviation principle with speed $m_Nk_N$ and rate function given by the relative entropy with respect to the standard normal distribution. Further, we complement the large deviation principle (LDP) obtained by Kabluchko and Prochno in [Large deviations for random matrices in the orthogonal group and Stiefel manifold with applications to random projections of product distributions, Annales de l'Institut Henri Poincaré. 60 (2024), 990 -- 1024] for the whole block $A_N$ with a moderate deviation principle (MDP). Concretely, we show an MDP for the sequence of matrices $β_N A_N$ in the product topology, where $β_N\to\infty$ is a sequence of real numbers such that $β_N = o(\sqrt{N})$. Here, in contrast to the LDP, the Gaussian behavior of the entries is reflected in the rate function. |
| title | Large and Moderate Deviations for Entries of Orthogonal Matrices and the Stiefel Manifold |
| topic | Probability Primary 60B20, 60F10, Secondary 15B52 |
| url | https://arxiv.org/abs/2509.24538 |