  Donsker-type functional limit theorems are proved for empirical processes
arising from discretely sampled increments of a univariate L\'evy process. In
the asymptotic regime the sampling frequencies increase to infinity and the
limiting object is a Gaussian process that can be obtained from the composition
of a Brownian motion with a covariance operator determined by the L\'evy
measure. The results are applied to derive the asymptotic distribution of
natural estimators for the distribution function of the L\'evy jump measure. As
an application we deduce Kolmogorov-Smirnov type tests and confidence bands.
