We develop two novel couplings between general pure-jump L\'evy processes in $\R^d$ and apply them to obtain upper bounds on the rate of convergence in an appropriate Wasserstein distance on the path space for a wide class of L\'evy processes attracted to a multidimensional stable process in the small-time regime. We also establish general lower bounds based on certain universal properties of slowly varying functions and the relationship between the Wasserstein and Toscani--Fourier distances of the marginals. Our upper and lower bounds typically have matching rates. In particular, the rate of convergence is polynomial for the domain of normal attraction and slower than a slowly varying function for the domain of non-normal attraction.

翻译：本文针对$\R^d$中的一般纯跳跃Lévy过程，构建了两种新型耦合方法，并将其应用于一大类在小时间区间内受多维稳定过程吸引的Lévy过程。我们在路径空间的适当Wasserstein距离下，获得了收敛速率的上界估计。同时，基于缓变函数的普适性质以及边缘分布的Wasserstein距离与Toscani-Fourier距离之间的关系，建立了相应的下界估计。所得上下界通常具有匹配的收敛速率。特别地，在正态吸引域中收敛速率为多项式阶，而在非正态吸引域中收敛速率慢于任意缓变函数。