On the regularity conditions in the CLT for the LUE

We consider the Laguerre Unitary Ensemble (LUE), the set of $n\times n$ sample covariance matrices $M = \frac{1}{n}X^*X$ where the $m\times n$ ($n \le m$) matrix $X$ has i.i.d. standard complex Gaussian entries. In particular we are concerned with the case where $\alpha := m - n$ is fixed, in which case the limiting eigenvalue density has a hard edge at $0$. We study the minimal regularity conditions required for a central limit theorem (CLT) type result to hold for the linear spectral statistics of the LUE. As long as the expression for the limiting variance is finite (and a slightly stonger condition holds near the soft edge) we show that the variance of the linear spectral statistic converges and consequently the CLT holds. Our methods are based on analyzing the explicit kernel for the LUE using asymptotics of the Laguerre polynomials. The CLT follows from approximating the test function of the statistic by Chebyshev polynomials.