Consider the product of m independent n×n random matrices from the spherical ensemble for m≥1. The empirical distribution based on the n eigenvalues of the product is called the empirical spectral distribution. Two recent papers by Götze, Kösters and Tikhomirov (2015) and Zeng (2016) obtain the limit of the empirical spectral distribution for the product when m is a fixed integer. In this paper, we investigate the limiting empirical distribution of scaled eigenvalues for the product of m independent matrices from the spherical ensemble in the case when m changes with n, that is, m=mn is an arbitrary sequence of positive integers.
Bibliographical noteFunding Information:
We would like to thank two reviewers for their constructive suggestions that have led to improvement in the layout and 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 (11171251), and the Major Program of Tianjin University of Finance and Economics (ZD1302).
© 2017 Elsevier B.V.
- Empirical spectral distribution
- Product ensemble
- Random matrix
- Spherical ensemble