Plot of CLT
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- by Tiefeng Jiang and Yutao Ma PDF
- Trans. Amer. Math. Soc. 372 (2019), 1509-1553 Request permission
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 $\sigma$.References
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Additional Information
- Tiefeng Jiang
- Affiliation: School of Statistics, University of Minnesota, 224 Church Street SE, Minneapolis, Minnesota 55455
- MR Author ID: 312116
- 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
- MR Author ID: 792222
- Email: mayt@bnu.edu.cn
- 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 - © Copyright 2019 American Mathematical Society
- 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
- MathSciNet review: 3976569