### 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) |
---|---|

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 |

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### Keywords

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

### Cite this

*Transactions of the American Mathematical Society*,

*372*(3), 1509-1553. https://doi.org/10.1090/tran/7470

**Distances between random orthogonal matrices and independent normals.** / Jiang, Tiefeng; Ma, Yutao.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 372, no. 3, pp. 1509-1553. https://doi.org/10.1090/tran/7470

}

TY - JOUR

T1 - Distances between random orthogonal matrices and independent normals

AU - Jiang, Tiefeng

AU - Ma, Yutao

PY - 2019/8/1

Y1 - 2019/8/1

N2 - 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.

AB - 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.

KW - Convergence of probability measure

KW - Haar measure

KW - Orthogonal group

KW - Random matrix

UR - http://www.scopus.com/inward/record.url?scp=85065407212&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85065407212&partnerID=8YFLogxK

U2 - 10.1090/tran/7470

DO - 10.1090/tran/7470

M3 - Article

AN - SCOPUS:85065407212

VL - 372

SP - 1509

EP - 1553

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 3

ER -