Motivated by the stochastic block model, we investigate a class of Wigner-type matrices with certain block structures and establish a CLT for the corresponding linear spectral statistics (LSS) via the large-deviation bounds from local law and the cumulant expansion formula. We apply the results to the stochastic block model. Specifically, a class of renormalized adjacency matrices will be block-Wigner-type matrices. Further, we show that for certain estimator of such renormalized adjacency matrices, which will be no longer Wigner-type but share long-range non-decaying weak correlations among the entries, the LSS of such estimators will still share the same limiting behavior as those of the block-Wigner-type matrices, thus enabling hypothesis testing about stochastic block model.

受随机块模型的启发，我们研究了一类具有一定块结构的 Wigner 型矩阵，并通过局部定律的大偏差界限和累积量展开公式为相应的线性谱统计 (LSS) 建立了 CLT。我们将结果应用于随机块模型。具体来说，一类重归一化的邻接矩阵将是分块维格纳型矩阵。此外，我们表明，对于此类重新归一化邻接矩阵的某些估计量（不再是 Wigner 型但在条目之间共享长程非衰减弱相关性），此类估计量的 LSS 仍将具有与那些估计量相同的限制行为块维格纳型矩阵，从而实现关于随机块模型的假设检验。