  Covariances and variances of linear statistics of a point process can be
written as integrals over the truncated two-point correlation function. When
the point process consists of the eigenvalues of a random matrix ensemble,
there are often large $N$ universal forms for this correlation after smoothing,
which results in particularly simple limiting formulas for the fluctuation of
the linear statistics. We review these limiting formulas, derived in the
simplest cases as corollaries of explicit knowledge of the truncated two-point
correlation. One of the large $N$ limits is to scale the eigenvalues so that
limiting support is compact, and the linear statistics vary on the scale of the
support. This is a global scaling. The other, where a thermodynamic limit is
first taken so that the spacing between eigenvalues is of order unity, and then
a scale imposed on the test functions so they are slowly varying, is the bulk
scaling. The latter was already identified as a probe of random matrix
characteristics for quantum spectra in the pioneering work of Dyson and Mehta.
