Empirical Spectral Distribution for Product of Spherical Ensembles

Consider the product of $m$ independent $n\times n$ spherical ensembles for $m\ge 1$. The empirical distribution based on the $n$ eigenvalues of the product is called the empirical spectral distribution. A recent paper by Zeng (2016) obtains the limit of the empirical spectral distribution for the product ensemble when $m$ is a fixed integer. In this paper, we investigate the limiting empirical spectral distribution for the product of spherical ensembles in the case when $m$ changes with $n$, that is, $m=m_n$ is an arbitrary sequence of positive integers.