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Transactions of the American Mathematical Society

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Distances between random orthogonal matrices and independent normals


Authors: Tiefeng Jiang and Yutao Ma
Journal: Trans. Amer. Math. Soc. 372 (2019), 1509-1553
MSC (2010): Primary 15B52, 28C10, 51F25, 60B15, 62E17
DOI: https://doi.org/10.1090/tran/7470
Published electronically: May 7, 2019
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Abstract: Let $ \bold {\Gamma }_n$ be an $ n\times n$ Haar-invariant orthogonal matrix. Let $ \mathbf Z_n$ be the $ p\times q$ upper-left submatrix of $ \bold {\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 $.


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Additional Information

Tiefeng Jiang
Affiliation: School of Statistics, University of Minnesota, 224 Church Street SE, Minneapolis, Minnesota 55455
Email: jiang040@umn.edu

Yutao Ma
Affiliation: School of Mathematical Sciences $&$ Laboratory of Mathematics and Complex Systems of Ministry of Education, Beijing Normal University, 100875 Beijing, People’s Republic of China
Email: mayt@bnu.edu.cn

DOI: https://doi.org/10.1090/tran/7470
Keywords: Haar measure, orthogonal group, random matrix, convergence of probability measure
Received by editor(s): April 17, 2017
Received by editor(s) in revised form: November 9, 2017
Published electronically: May 7, 2019
Additional Notes: The research of the first author was supported in part by NSF Grants DMS-1209166 and DMS-1406279.
The research of the second author was supported in part by NSFC 11431014, 11371283, 11571043 and 985 Projects.
Tiefeng Jiang is the corresponding author
Article copyright: © Copyright 2019 American Mathematical Society