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

TL;DR

This paper establishes a central limit theorem for linear spectral statistics of block-structured Wigner-type matrices, with applications to stochastic block models and hypothesis testing.

Contribution

It introduces a CLT for a class of block-structured Wigner matrices and extends the results to estimators with long-range correlations, aiding hypothesis testing in stochastic block models.

Findings

CLT for block-Wigner-type matrices established.

Linear spectral statistics share the same limit for certain estimators.

Applications to hypothesis testing in stochastic block models.

Abstract

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 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 linear spectral statistics 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.