Plot of CLT

Abstract:

Let $\mathbf {\Gamma }_n$ be an $n\times n$ Haar-invariant orthogonal matrix. Let $\mathbf Z_n$ be the $p\times q$ upper-left submatrix of $\mathbf {\Gamma }_n,$ where $p=p_n$ and $q=q_n$ are two positive integers. Let $\mathbf G_n$ be a $p\times q$ matrix whose $pq$ entries are independent standard normals. In this paper we consider the distance between $\sqrt {n}\mathbf Z_n$ and $\mathbf G_n$ in terms of the total variation distance, the Kullback-Leibler distance, the Hellinger distance, and the Euclidean distance. We prove that each of the first three distances goes to zero as long as $pq/n$ goes to zero, and not so if $(p, q)$ sits on the curve $pq=\sigma n$, where $\sigma$ is a constant. However, it is different for the Euclidean distance, which goes to zero provided $pq^2/n$ goes to zero, and not so if $(p,q)$ sits on the curve $pq^2=\sigma n.$ A previous work by Jiang (2006) shows that the total variation distance goes to zero if both $p/\sqrt {n}$ and $q/\sqrt {n}$ go to zero, and it is not true provided $p=c\sqrt {n}$ and $q=d\sqrt {n}$ with $c$ and $d$ being constants. One of the above results confirms a conjecture that the total variation distance goes to zero as long as $pq/n\to 0$ and the distance does not go to zero if $pq=\sigma n$ for some constant $\sigma$.