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.
- Random matrices
- Heavy tailed random variables
- Central limit theorem