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

Račkauskas, Alfredas; Suquet, Charles

ESAIM: Probability and Statistics, Tome 24 (2020), pp. 186-206.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

Let ξ$$ be the polygonal line partial sums process built on i.i.d. centered random variables X$$, i ≥ 1. The Bernstein-Kantorovich theorem states the equivalence between the finiteness of E|X₁|$$ and the joint weak convergence in C[0, 1] of n^−1∕2ξ$$ to a Brownian motion W with the moments convergence of $$ to $$. For 0 < α < 1∕2 and p (α) = (1 ∕ 2 - α) ^-1, we prove that the joint convergence in the separable Hölder space $$ of n^−1∕2ξ$$ to W jointly with the one of $$ to $$ holds if and only if P(|X₁| > t) = o(t$$) when r < p(α) or E|X₁|$$ < ∞ 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|$$ < ∞. In the case where the X$$’s are p regularly varying, we can complete these results for α > 1∕2 − 1∕p with an appropriate normalization.

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