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Consider the product of $m$ independent $n\times n$ random matrices from thespherical ensemble for $m\ge 1$. The spectral radius is defined as the maximumabsolute value of the $n$ eigenvalues of the product matrix. When $m=1$, thelimiting distribution for the spectral radii has been obtained by Jiang and Qi(2017). In this paper, we investigate the limiting distributions for thespectral radii in general. When $m$ is a fixed integer, we show that thespectral radii converge weakly to distributions of functions of independentGamma 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.

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