Fisher information approximation of random orthogonal matrices by Gaussian matrices

Yutong Chen; Yutao Ma; Shuhong Xie; Zhuoya Yao

paper

Fisher information approximation of random orthogonal matrices by Gaussian matrices

arXiv:2504.10887

Abstract

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 $L (n Z_{n})$ and $L (G_{n})$ be their joint distribution, respectively. Consider the Fisher information $I (L (n Z_{n}) ∣ L (G_{n}))$ between the distributions of $n Z_{n}$ and $G_{n} .$ In this paper, we conclude that $I (L (n Z_{n}) ∣ L (G_{n})) ⟶ 0$ as $n \to \infty$ if $pq = o (n)$ and it does not tend to zero if $c = n \to \infty lim \frac{pq}{n} \in (0, + \infty) .$ Precisely, we obtain that $I (L (n Z_{n}) ∣ L (G_{n})) = \frac{p ^{2} q ( q + 1 )}{4 n ^{2}} (1 + o (1))$ when $p = o (n) .$

18 pages