The Weighted Bootstrap Mean for Heavy-Tailed Distributions

We study the performance of the weighted bootstrap of the mean of i.i.d. random variables, X ₁, X ₂,..., in the domain of attraction of an -stable law, 1<<2. In agreement with the results, in the Efron's bootstrap setup, by Athreya,⁽⁴⁾ Arcones and Giné⁽²⁾ and Deheuvels et al.,⁽¹¹⁾ we prove that for a low resampling intensity the weighted bootstrap works in probability. Our proof resorts to the 0–1 law methodology introduced in Arenal and Matrán⁽³⁾. This alternative to the methodology initiated in Mason and Newton⁽²⁵⁾ presents the advantage that it does not use Hájek's Central 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 Wasserstein distance between the empirical and the parent distribution of a sample, which is also a main tool in our development.