A classical result in random matrix theory reveals that the limiting spectral distribution of a Wigner matrix whose entries have a common variance and satisfy other regular assumptions almost surely converges to the semicircular law. In the paper, we will relax the assumption of uniform covariance of each entry, when the average of the normalized sums of the variances in each row of the data matrix converges to a constant, we prove that the same limiting spectral distribution holds. A similar result on a sample covariance matrix is also established. The proofs mainly depend on the Stein equation and the generalized Stein equation of independent random variables.

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方差条目不相等的随机矩阵的极限谱分布的结果

随机矩阵理论的经典结果表明，维格纳矩阵的极限光谱分布其条目具有共同方差并满足其他正则假设，几乎可以肯定地收敛于半圆定律。在本文中，我们将放宽每个条目的均匀协方差的假设，当数据矩阵每一行中的方差的归一化总和的平均值收敛为常数时，我们证明了相同的极限频谱分布成立。还建立了样本协方差矩阵的相似结果。证明主要依赖于Stein方程和独立随机变量的广义Stein方程。