We consider random Hermitian matrices with independent upper triangular entries. Wigner’s semicircle law says that under certain additional assumptions, the empirical spectral distribution converges to the semicircle distribution. We characterize convergence to semicircle in terms of the variances of the entries, under natural assumptions such as the Lindeberg condition. The result extends to certain matrices with entries having infinite second moments. As a corollary, another characterization of semicircle convergence is given in terms of convergence in distribution of the row sums to the standard normal distribution.

中文翻译：

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收敛于半圆分布的充要条件

我们考虑具有独立上三角项的随机 Hermitian 矩阵。维格纳半圆定律说，在某些附加假设下，经验光谱分布收敛于半圆分布。在诸如 Lindeberg 条件之类的自然假设下，我们根据条目的方差来描述收敛到半圆的特征。结果扩展到某些矩阵，其条目具有无限的二阶矩。作为推论，半圆收敛的另一个特征是根据行总和的分布收敛到标准正态分布给出的。