On Bernstein–Kantorovich invariance principle in Hölder spaces and weighted scan statistics^*

¹ Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania.
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

Received: 10 July 2018
Accepted: 14 November 2019

Abstract

Let ξ_n be the polygonal line partial sums process built on i.i.d. centered random variables X_i, 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∕2ξ_n to a Brownian motion W with the moments convergence of E∥_n^−1/2ξ_n∥_∞^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 H_α^o of n^−1∕2ξ_n to W jointly with the one of E∥_n^−1∕2ξ_n∥_α^r to E∥W∥_α^r 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 X_i’s are p regularly varying, we can complete these results for α > 1∕2 − 1∕p with an appropriate normalization.