## Abstract

Let Γ_{n} be an n × n Haar-invariant orthogonal matrix. Let Z_{n} be the p × q upper-left submatrix of Γ_{n}, where p = p_{n} and q = q_{n} are two positive integers. Let G_{n} be a p × q matrix whose pq entries are independent standard normals. In this paper we consider the distance between^{√}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/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 pq^{2}/n goes to zero, and not so if (p, q) sitsonthecurvepq^{2} = 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

Funding Information: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.

Funding Information:

Received by the editors April 17, 2017, and, in revised form, November 9, 2017. 2010 Mathematics Subject Classification. Primary 15B52, 28C10, 51F25, 60B15, 62E17. Key words and phrases. Haar measure, orthogonal group, random matrix, convergence of probability measure. 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.

Publisher Copyright:

© 2019 American Mathematical Society.

## Keywords

- Convergence of probability measure
- Haar measure
- Orthogonal group
- Random matrix