Fisher information approximation of random orthogonal matrices by Gaussian matrices

Let [Formula: see text] be an [Formula: see text] Haar-invariant orthogonal matrix. Let [Formula: see text] be the [Formula: see text] upper-left submatrix of [Formula: see text] and [Formula: see text] be a [Formula: see text] matrix whose [Formula: see text] entries are independent standard normals, where [Formula: see text] and [Formula: see text] are two positive integers. Let [Formula: see text] and [Formula: see text] be their joint distributions, respectively. Consider the Fisher information [Formula: see text] between the distributions of [Formula: see text] and [Formula: see text] In this paper, we conclude that [Formula: see text] as [Formula: see text] if [Formula: see text] and it does not tend to zero if [Formula: see text] Precisely, we obtain that [Formula: see text] when [Formula: see text]