Generally, a statistic of a point process refers to the stochastic variable defined by evaluating a function at the points. In this setting the class of functions of a single variable are referred to as linear statistics. One example is the counting function for eigenvalues in a specified region. The large region size form of the variance of this statistic, when proportional to the surface area, is used to specify the state as being hyperuniform. In the case of GinUE eigenvalues, a central limit theorem for the distribution can be established, and this can be strengthened for a local limit theorem of the underlying probabilities. Moreover, large deviation formulas for the latter are obtained. The counting function linear statistic is discontinuous. Smooth linear statistics with global scaling exhibit distinct large N behaviours, in particular the variance is now an O(1) quantity for GinUE eigenvalues. A decomposition of this variance as a contribution from the bulk, and a contribution from the boundary, is exhibited. The chapter concludes with a discussion of GinUE eigenvalues in spatial modelling.