Abstract
Let Γn be an n × n Haar-invariant orthogonal matrix. Let Zn be the p × q upper-left submatrix of Γn, where p = pn and q = qn are two positive integers. Let Gn be a p × q matrix whose pq entries are independent standard normals. In this paper we consider the distance between√nZn and Gn 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 = σn, whereσ is a constant. However, it is different for the Euclidean distance, which goes to zero provided pq2/n goes to zero, and not so if (p, q) sitsonthecurvepq2 = σn. A previous work by Jiang (2006) shows that the total variation distance goes to zero if both p/√n and q/√n go to zero, and it is not true provided p = c√n and q = d√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 = σn for some constant σ.
Original language | English (US) |
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Pages (from-to) | 1509-1553 |
Number of pages | 45 |
Journal | Transactions of the American Mathematical Society |
Volume | 372 |
Issue number | 3 |
DOIs | |
State | Published - Aug 1 2019 |
Bibliographical note
Publisher Copyright:© 2019 American Mathematical Society.
Keywords
- Convergence of probability measure
- Haar measure
- Orthogonal group
- Random matrix