High-frequency Donsker theorems for Lévy measures

Given empirical measures $P_{\Delta,n}$ arising from increments $(L_{k\Delta}-L_{(k-1)\Delta}) \sim P_\Delta$, $k=1,...,n$, of a univariate L\'evy process sampled discretely at frequency $\Delta \to 0$, we prove Donsker-type functional limit theorems for suitably scaled random measures $(P_{\Delta,n}-P_\Delta) \ast F^{-1}m$, where $F^{-1}m$ is a general sequence of approximate identities. Examples include both classical empirical processes ($m=1$) and a flexible family of smoothed empirical processes. The limiting random variable is Gaussian and can be obtained from the composition of a Brownian motion with a covariance operator determined by the L\'evy measure $\nu$ of the process. The convolution $P_\Delta \ast F^{-1}m$ is a natural object in the asymptotic identification of $\nu$ and appears both in a direct estimation approach and in an approach based on the empirical characteristic function. We deduce several applications to statistical inference on the distribution function of $\nu$ based on discrete high frequency samples of the L\'evy process, including Kolmogorov-Smirnov type limit theorems.