Uniform convergence of the empirical spectral distribution function

Let X be a linear process having a finite fourth moment. Assume F is a class of square-integrable functions. We consider the empirical spectral distribution function Jn,X based on X and indexed by F. If F is totally bounded then Jn,X satisfies a uniform strong law of large numbers. If, in addition, a metric entropy condition holds, then Jn,X obeys the uniform central limit theorem.