  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.
