Fluctuations of linear statistics of half-heavy-tailed random matrices

In this paper, we consider a Wigner matrix A with entries whose cumulative distribution decays as x−α with 2<α<4 for large x. We are interested in the fluctuations of the linear statistics N−1Trφ(A), for some nice test functions φ. The behavior of such fluctuations has been understood for both heavy-tailed matrices (i.e. α<2) in Benaych-Georges (2014) and light-tailed matrices (i.e. α>4) in Bai and Silverstein (2009). This paper fills in the gap of understanding it for 2<α<4. We find that while linear spectral statistics for heavy-tailed matrices have fluctuations of order N−1/2 and those for light-tailed matrices have fluctuations of order N−1, the linear spectral statistics for half-heavy-tailed matrices exhibit an intermediate α-dependent order of N−α/4.