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

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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.