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^-1 Trφ(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.