Existing results for the estimation of the L\'evy measure are mostly limited to the onedimensional setting. We apply the spectral method to multidimensional L\'evy processes in order to construct a nonparametric estimator for the multivariate jump distribution. We prove convergence rates for the uniform estimation error under both a low- and a high-frequency observation regime. The method is robust to various dependence structures. Along the way, we present a uniform risk bound for the multivariate empirical characteristic function and its partial derivatives. The method is illustrated with simulation examples.

翻译：现有关于Lévy测度估计的研究结果大多局限于单维情形。本文将谱方法应用于多维Lévy过程，以构建多变量跳跃分布的非参数估计量。我们在低频和高频观测机制下验证了均匀估计误差的收敛速度。该方法对多种依赖结构具有稳健性。在此过程中，我们给出了多变量经验特征函数及其偏导数的均匀风险界。通过模拟算例对所提方法进行了说明。