Volume 24, 2020
|Page(s)||186 - 206|
|Published online||06 March 2020|
On Bernstein–Kantorovich invariance principle in Hölder spaces and weighted scan statistics*
Department of Mathematics and Informatics, Vilnius University,
2 Laboratoire P. Painlevé (UMR 8524 CNRS), Bât. M2, Cité Scientifique, Université de Lille, 59655 Villeneuve d’Ascq Cedex, France.
** Corresponding author: Charles.Suquet@univ-lille.fr
Accepted: 14 November 2019
Let ξn be the polygonal line partial sums process built on i.i.d. centered random variables Xi, i ≥ 1. The Bernstein-Kantorovich theorem states the equivalence between the finiteness of E|X1|max(2,r) and the joint weak convergence in C[0, 1] of n−1∕2ξn to a Brownian motion W with the moments convergence of E∥n−1/2ξn∥∞r to E∥W∥∞r. For 0 < α < 1∕2 and p (α) = (1 ∕ 2 - α) -1, we prove that the joint convergence in the separable Hölder space Hαo of n−1∕2ξn to W jointly with the one of E∥n−1∕2ξn∥αr to E∥W∥αr holds if and only if P(|X1| > t) = o(t−p(α)) when r < p(α) or E|X1|r < ∞ when r ≥ p(α). As an application we show that for every α < 1∕2, all the α-Hölderian moments of the polygonal uniform quantile process converge to the corresponding ones of a Brownian bridge. We also obtain the asymptotic behavior of the rth moments of some α-Hölderian weighted scan statistics where the natural border for α is 1∕2 − 1∕p when E|X1|p < ∞. In the case where the Xi’s are p regularly varying, we can complete these results for α > 1∕2 − 1∕p with an appropriate normalization.
Mathematics Subject Classification: 60F17 / 62G30
Key words: Fonctional central limit theorem / Hölder space / moments / quantile process / regular variation / scan statistics / Wasserstein distance
© The authors. Published by EDP Sciences, SMAI 2020
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