Limiting Distributions of Spectral Radii for Product of Matrices from the Spherical Ensemble

Consider the product of $m$ independent $n\times n$ random matrices from the spherical ensemble for $m\ge 1$. 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.