Abstract
In the present paper, we prove that under the assumption of the finite sixth moment for elements of a Wigner matrix, the convergence rate of its empirical spectral distribution to the Wigner semicircular law in probability is $O(n^{-1/2})$ when the dimension n tends to infinity.
Highlights
Introduction and the resultA Wigner matrix Wn = n−1/2ni,j=1 is defined to be a Hermitian random matrix whose entries on and above the diagonal are independent zero-mean random variables
It is an important model for depicting heavy-nuclei atoms, which begin with the seminal work of Wigner in 1955 ([15])
The rate of convergence is important in establishing the central limit theorem for linear spectral statistics of Wigner matrices ([7, 6])
Summary
It is proved that,under assumptions of for all i, j, E|xij|2 = σ2, the ESD F Wn(x) converges almost surely to a non-random distribution F (x) which has the destiny function Convergence rate, Wigner matrix, Semicircular Law, spectral distribution. The rate of convergence is important in establishing the central limit theorem for linear spectral statistics of Wigner matrices ([7, 6]). In [2], Bai proved that under the assumption of supn supi,j Ex4ij < ∞, the rate of
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