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Motivated by the stochastic block model, we investigate a class ofWigner-type matrices with certain block structures, and establish a CLT for thecorresponding linear spectral statistics via the large-deviation bounds fromlocal law and the cumulant expansion formula. We apply the results to thestochastic block model. Specifically, a class of renormalized adjacencymatrices will be block-Wigner-type matrices. Further, we show that for certainestimator of such renormalized adjacency matrices, which will be no longerWigner-type but share long-range non-decaying weak correlations among theentries, the linear spectral statistics of such estimators will still share thesame limiting behavior as those of the block-Wigner-type matrices, thusenabling hypothesis testing about stochastic block model.

Central limit theorem for linear spectral statistics of block-Wigner-type matrices