## Abstract

Consider a truncated circular unitary matrix which is a p_{n} by p_{n} submatrix of an n by n circular unitary matrix by deleting the last n−p_{n} columns and rows. Jiang and Qi [11] proved that the maximum absolute value of the eigenvalues (known as spectral radius) of the truncated matrix, after properly normalized, converges in distribution to the Gumbel distribution if p_{n}/n is bounded away from 0 and 1. In this paper we investigate the limiting distribution of the spectral radius under one of the following four conditions: (1). p_{n}→∞ and p_{n}/n→0 as n→∞; (2). (n−p_{n})/n→0 and (n−p_{n})/(logn)^{3}→∞ as n→∞; (3). n−p_{n}→∞ and (n−p_{n})/logn→0 as n→∞ and (4). n−p_{n}=k≥1 is a fixed integer. We prove that the spectral radius converges in distribution to the Gumbel distribution under the first three conditions and to a reversed Weibull distribution under the fourth condition.

Original language | English (US) |
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Pages (from-to) | 536-554 |

Number of pages | 19 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 458 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2018 |

### Bibliographical note

Funding Information:We would like to thank an anonymous referee for his/her careful reading of the original version of the paper and pointing out some imperfections in the proofs. Gui's work was partially supported by the program for the Fundamental Research Funds for the Central Universities ( 2014RC042 ).

Publisher Copyright:

© 2017 Elsevier Inc.

## Keywords

- Circular unitary matrix
- Eigenvalue
- Extreme value
- Limiting distribution
- Spectral radius