Optimal Bounds for Convergence of Expected Spectral Distributions to the Semi-Circular Law

2014/05/30 by Götze, F., Tikhomirov, A.
#15B52 #60B20 #FOS: Mathematics #Probability (math.PR) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1405.7820

Abstract

Let \mathbf X=(X_jk)_j,k=1ⁿ denote a Hermitian random matrix with entries X_jk, which are independent for 1≤ j≤ k≤ n. We consider the rate of convergence of the empirical spectral distribution function of the matrix \mathbf X to the semi-circular law assuming that \mathbf E X_jk=0, \mathbf E X_jk²=1 and that sup_n≥1sup_{1≤ j,k≤ n}\mathbf E|X_jk|⁴=:μ₄

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