The weighted bootstrap mean for heavy-tailed distributions

We study thc performance of the weighted bootstrap of the mean of i.i.d. ra ndom variables, X-1, X-2 ,..., in the domain of attraction of an a-stable l aw, 1 < alpha < 2. In agreement with the results, in the Efron's bootstrap setup, by Athreya,((4)) Arcones and Gine((2)) and Deheuvels et al.,((11)) w e prove that for a "low resampling intensity" the weighted bootstrap works in probability. Our proof results to the 0 1 law methodology introduced in Arenal and Matran((3)) This alternative to the methodology initiated in Mas on and Newton((25)) presents the advantage that it does not use Hajek's Cen tral Limit Theorem for linear rank statistics which actually only provides normal limit laws. We include as an appendix a sketched proof. based on the Komlos-Major Tusnady construction, of the asymptotic behaviour of the Wass erstein distance between the empirical and the parent distribution of ii sa mple, which is also a main tool in our development.