Nonparametric inference on Lévy measures and copulas

Abstract

In this paper nonparametric methods to assess the multivariate Lévy measure are introduced. Starting from high-frequency observations of a Lévy process $\mathbf{X} $, we construct estimators for its tail integrals and the Pareto–Lévy copula and prove weak convergence of these estimators in certain function spaces. Given $n$ observations of increments over intervals of length $\Delta_{n}$, the rate of convergence is $k_{n}^{-1/2}$ for $k_{n}=n\Delta_{n}$ which is natural concerning inference on the Lévy measure. Besides extensions to nonequidistant sampling schemes analytic properties of the Pareto–Lévy copula which, to the best of our knowledge, have not been mentioned before in the literature are provided as well. We conclude with a short simulation study on the performance of our estimators and apply them to real data.