Nonparametric inference on Lévy measures and copulas

Abstract

In this paper nonparametric methods to assess the multivariate Levy measure are introduced. Starting from high-frequency observations of a Levy process X, we construct estimators for its tail integrals and the Pareto Levy copula and prove weak convergence of these estimators in certain function spaces. Given n observations of increments over intervals of length n, the rate of convergence is k1=2 n for kn = nn which is natural concerning inference on the Levy measure. Analytic properties of the Pareto Levy 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.