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(eng): In this paper, we aim to explore the speed of convergence of the Wasserstein distance between stable cumulative distribution functions and their empirical counterparts. The theoretical results are compared with the results provided by simulations. The need to use simulations is explained by the fact that all the theoretical results which relate to the speed of convergence of the Wasserstein Metric in the set-up of stable distributions are asymptotic; therefore, the question of when that theory starts to be valid remains open. The asymptotic results are true only for relatively large numbers of observations exceeding hundreds of thousands. In cases dealing with lower numbers of observations, the speed of convergence turns out to be much slower than we might expect.