Fisher information approximation of random orthogonal matrices by Gaussian matrices

Let ${\mathrm{\Gamma }}_n$ be an $n\times n$ Haar-invariant orthogonal matrix. Let $Z_n$ be the $p\times q$ upper-left submatrix of ${\mathrm{\Gamma }}_n$ 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 $I(\mathcal{L}(\sqrt{n}{Z}_n)\mathrm{\mid }\mathcal{L}(G_n))$ between the distributions of $\sqrt{n}{Z}_n$ and $G_n\mathrm{.}$ In this paper, we conclude that $$I(\mathcal{L}(\sqrt{n}{Z}_n)\mathrm{\mid }\mathcal{L}(G_n))⟶0$$ as $n\to \mathrm{\infty }$ if $pq=o(n)$ and it does not tend to zero if $c=\underset{n\to \mathrm{\infty }}{\lim }\frac{pq}{n}\in (0,+\mathrm{\infty })\mathrm{.}$ Precisely, we obtain that $$I(\mathcal{L}(\sqrt{n}{Z}_n)\mathrm{\mid }\mathcal{L}(G_n))=\frac{p^{2}q(q+1)}{4n^2}(1+o(1))$$ when $p=o(n)\mathrm{.}$