  We study the behavior of the empirical distribution function of iterates of
intermittent maps in the Hilbert space of square inegrable functions with
respect to Lebesgue measure. In the long-range dependent case, we prove that
the empirical distribution function, suitably normalized, converges to a
degenerate stable process, and we give the corresponding almost sure result. We
apply the results to the convergence of the Wasserstein distance between the
empirical measure and the invariant measure. We also apply it to obtain the
asymptotic distribution of the corresponding Cram\'er-von-Mises statistic.
