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.
Bibliographical noteFunding Information:
We would like to thank the reviewer whose constructive suggestions have led to improvement in the readability of the paper. Chang's research was supported in part by the Major Research Plan of the National Natural Science Foundation of China ( 91430108 ), the National Basic Research Program ( 2012CB955804 ), the National Natural Science Foundation of China ( 11771322 ), and the Major Program of Tianjin University of Finance and Economics ( ZD1302 ). Li's research was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada (Grant #: RGPIN-2014-05428 ).
- Limiting distribution
- Product ensemble
- Random matrix
- Spectral radius
- Spherical ensemble