Limiting distributions of spectral radii for product of matrices from the spherical ensemble

Shuhua Chang, Deli Li, Yongcheng Qi

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Abstract

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.

Original language English (US) 1165-1176 12 Journal of Mathematical Analysis and Applications 461 2 https://doi.org/10.1016/j.jmaa.2018.01.048 Published - May 15 2018

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Limiting Distribution
Ensemble
Random variables
Infinity
Matrix Product
Random Matrices
Absolute value
Logarithmic
Random variable
Tend
Eigenvalue
Converge
Integer

Keywords

• Limiting distribution
• Product ensemble
• Random matrix
• Spherical ensemble

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In: Journal of Mathematical Analysis and Applications, Vol. 461, No. 2, 15.05.2018, p. 1165-1176.

Research output: Contribution to journalArticle

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