### Abstract

Consider the product of m independent n×n random matrices from the spherical ensemble for m≥1. The spectral radius is defined as the maximum absolute value of the n eigenvalues of the product matrix. When m=1, the limiting distribution for the spectral radii has been obtained by Jiang and Qi (2017). In this paper, we investigate the limiting distributions for the spectral radii in general. When m is a fixed integer, we show that the spectral radii converge weakly to distributions of functions of independent Gamma random variables. When m=m_{n} tends to infinity as n goes to infinity, we show that the logarithmic spectral radii have a normal limit.

Original language | English (US) |
---|---|

Pages (from-to) | 1165-1176 |

Number of pages | 12 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 461 |

Issue number | 2 |

DOIs | |

State | Published - May 15 2018 |

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### Keywords

- Limiting distribution
- Product ensemble
- Random matrix
- Spectral radius
- Spherical ensemble

### Cite this

*Journal of Mathematical Analysis and Applications*,

*461*(2), 1165-1176. https://doi.org/10.1016/j.jmaa.2018.01.048

**Limiting distributions of spectral radii for product of matrices from the spherical ensemble.** / Chang, Shuhua; Li, Deli; Qi, Yongcheng.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 461, no. 2, pp. 1165-1176. https://doi.org/10.1016/j.jmaa.2018.01.048

}

TY - JOUR

T1 - Limiting distributions of spectral radii for product of matrices from the spherical ensemble

AU - Chang, Shuhua

AU - Li, Deli

AU - Qi, Yongcheng

PY - 2018/5/15

Y1 - 2018/5/15

N2 - Consider the product of m independent n×n random matrices from the spherical ensemble for m≥1. The spectral radius is defined as the maximum absolute value of the n eigenvalues of the product matrix. When m=1, the limiting distribution for the spectral radii has been obtained by Jiang and Qi (2017). In this paper, we investigate the limiting distributions for the spectral radii in general. When m is a fixed integer, we show that the spectral radii converge weakly to distributions of functions of independent Gamma random variables. When m=mn tends to infinity as n goes to infinity, we show that the logarithmic spectral radii have a normal limit.

AB - Consider the product of m independent n×n random matrices from the spherical ensemble for m≥1. The spectral radius is defined as the maximum absolute value of the n eigenvalues of the product matrix. When m=1, the limiting distribution for the spectral radii has been obtained by Jiang and Qi (2017). In this paper, we investigate the limiting distributions for the spectral radii in general. When m is a fixed integer, we show that the spectral radii converge weakly to distributions of functions of independent Gamma random variables. When m=mn tends to infinity as n goes to infinity, we show that the logarithmic spectral radii have a normal limit.

KW - Limiting distribution

KW - Product ensemble

KW - Random matrix

KW - Spectral radius

KW - Spherical ensemble

UR - http://www.scopus.com/inward/record.url?scp=85044353532&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85044353532&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2018.01.048

DO - 10.1016/j.jmaa.2018.01.048

M3 - Article

VL - 461

SP - 1165

EP - 1176

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -