On Non-parametric Estimation of the Lévy Kernel of Recurrent Markov Processes

We observe a sample path of a multivariate strong Markov process and, in particular, all its jumps up to some time t. The law of the jumps is described by the L\'evy kernel. We construct estimators for the density of the L\'evy kernel and prove consistency as well as a central limit theorem as t tends to infinity. In the positive recurrent case, our estimator is asymptotically normal; in the null recurrent case, our estimator is asymptotically mixed normal. The rate of convergence in the central limit theorem is arbitrarily close to the non-parametric minimax rate of smooth density estimation. Asymptotic confidence intervals are provided.