Rate of Convergence of the Empirical Spectral Distribution Function to the Semi-Circular Law

2014/07/10 by Götze, F., Tikhomirov, A. N.
#15B52 #60B20 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1407.2780

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 W=\frac1√ n\mathbf X to the semi-circular law assuming that \mathbf E X_jk=0, \mathbf E X_jk²=1 and uniformly bounded eight moments. By means of a recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the Wigner matrix \mathbf W and the semi--circular law is of order O(n^-1log⁵n) with high probability.

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