Fisher information approximation of random orthogonal matrices by Gaussian matrices
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913794890924032 |
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| author | Chen, Yutong Ma, Yutao Xie, Shuhong Yao, Zhuoya |
| author_facet | Chen, Yutong Ma, Yutao Xie, Shuhong Yao, Zhuoya |
| contents | Let $Γ_n$ be an $n\times n$ Haar-invariant orthogonal matrix. Let ${ Z}_n$ be the $p\times q$ upper-left submatrix of $Γ_n$ and ${G}_n$ be a $p\times q$ matrix whose $pq$ entries are independent standard normals, where $p$ and $q$ are two positive integers. Let $\mathcal{L}(\sqrt{n} {Z}_n)$ and $\mathcal{L}({G}_n)$ be their joint distribution, respectively. Consider the Fisher information $I(\mathcal{L}(\sqrt{n} { Z}_n)|\mathcal{L}(G_n))$ between the distributions of $\sqrt{n} {Z}_n$ and ${ G}_n.$ In this paper, we conclude that $$I(\mathcal{L}(\sqrt{n} {Z}_n)|\mathcal{L}(G_n))\longrightarrow 0 $$ as $n\to\infty$ if $pq=o(n)$ and it does not tend to zero if $c=\lim\limits_{n\to\infty}\frac{pq}{n}\in(0, +\infty).$ Precisely, we obtain that $$I(\mathcal{L}(\sqrt{n} {Z}_n)|\mathcal{L}(G_n))=\frac{p^2q(q+1)}{4n^2}(1+o(1))$$ when $p=o(n).$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_10887 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fisher information approximation of random orthogonal matrices by Gaussian matrices Chen, Yutong Ma, Yutao Xie, Shuhong Yao, Zhuoya Probability 60B20, 62B10, 62E17, 15B52 Let $Γ_n$ be an $n\times n$ Haar-invariant orthogonal matrix. Let ${ Z}_n$ be the $p\times q$ upper-left submatrix of $Γ_n$ and ${G}_n$ be a $p\times q$ matrix whose $pq$ entries are independent standard normals, where $p$ and $q$ are two positive integers. Let $\mathcal{L}(\sqrt{n} {Z}_n)$ and $\mathcal{L}({G}_n)$ be their joint distribution, respectively. Consider the Fisher information $I(\mathcal{L}(\sqrt{n} { Z}_n)|\mathcal{L}(G_n))$ between the distributions of $\sqrt{n} {Z}_n$ and ${ G}_n.$ In this paper, we conclude that $$I(\mathcal{L}(\sqrt{n} {Z}_n)|\mathcal{L}(G_n))\longrightarrow 0 $$ as $n\to\infty$ if $pq=o(n)$ and it does not tend to zero if $c=\lim\limits_{n\to\infty}\frac{pq}{n}\in(0, +\infty).$ Precisely, we obtain that $$I(\mathcal{L}(\sqrt{n} {Z}_n)|\mathcal{L}(G_n))=\frac{p^2q(q+1)}{4n^2}(1+o(1))$$ when $p=o(n).$ |
| title | Fisher information approximation of random orthogonal matrices by Gaussian matrices |
| topic | Probability 60B20, 62B10, 62E17, 15B52 |
| url | https://arxiv.org/abs/2504.10887 |