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

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| X 1 | max(2, r ) and the joint weak convergence in C [0, 1] of n −1∕2 ξ n to a Brownian motion W with the moments convergence of E∥ n −1/2 ξ 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∕2 ξ n to W jointly with the one of E∥ n −1∕2 ξ n ∥ α r to E∥ W ∥ α r holds if and only if P (| X 1 | > t ) = o ( t − p ( α ) ) when r < p ( α ) or E| X 1 | 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 r th moments of some α -Hölderian weighted scan statistics where the natural border for α is 1∕2 − 1∕ p when E| X 1 | 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. ; 24