DISTANCES BETWEEN RANDOM ORTHOGONAL MATRICES AND INDEPENDENT NORMALS
DISTANCES BETWEEN RANDOM ORTHOGONAL MATRICES AND INDEPENDENT NORMALS
复制标题
随机正交矩阵与独立法线之间的距离
DOI：
10.1090/tran/7470
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发表时间：
2019-08-01
影响因子：
1.3
通讯作者：
Ma, Yutao
中科院分区：
文献类型：
--
作者：
Jiang, Tiefeng;Ma, Yutao
Let Gamma(n) be an n x n Haar-invariant orthogonal matrix. Let Z(n) be the p x q upper-left submatrix of Gamma(n), where p = p(n) and q = q(n) are two positive integers. Let G(n) be a p x q matrix whose pq entries are independent standard normals. In this paper we consider the distance between root nZ(n) and 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 In 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 p(q2)/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/root n and q root n, go to zero, and it is not true provided p = c root n, and q = d root 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 -> 0 and the distance does not go to zero if pq = sigma n for some constant sigma.
