Hölderian invariance principle for hilbertian linear processes

Račkauskas, Alfredas; Suquet, Charles

doi:10.1051/ps:2008011

Račkauskas, Alfredas ; Suquet, Charles

ESAIM: Probability and Statistics, Tome 13 (2009), pp. 261-275.

Résumé
Abstract

Soit ${(ξ_{n})}_{n \geq 1}$ le processus polygonal de sommes partielles bâti sur le processus linéaire $X_{n} = \sum_{i \geq 0} a_{i} (ϵ_{n - i})$ , $n \geq 1$ , les ${(ϵ_{i})}_{i \in ℤ}$ étant des éléments aléatoires i.i.d., centrés d’un espace de Hilbert séparable $ℍ$ et les $a_{i}$ ’s des opérateurs linéaires bornés $ℍ \to ℍ$ , vérifiant $\sum_{i \geq 0} ∥ a_{i} ∥ < \infty$ . Nous étudions le théorème limite central fonctionnel pour $ξ_{n}$ dans les espaces de Hölder $H_{ρ}^{o} (ℍ)$ de fonctions $x : [0, 1] \to ℍ$ vérifiant $∥ x (t + h) - x (t) ∥ = o (ρ (h))$ uniformément en $t$ , où $ρ (h) = h^{α} L (1 / h)$ , $0 \leq h \leq 1$ avec $0 < α \leq 1 / 2$ et $L$ à variation lente. Nous prouvons la convergence en loi dans $H_{ρ}^{o} (ℍ)$ de $ξ_{n}$ vers un mouvement brownien à valeurs dans $ℍ$ , sous la condition optimale que pour tout $c > 0$ , $t P (∥ ϵ_{0} ∥ > c t^{1 / 2} ρ (1 / t)) = o (1)$ quand $t$ tend vers l’infini, au prix dans le cas limite $α = 1 / 2$ d’une légère restriction sur $L$ . Notre résultat s’applique en particulier au cas $ρ (h) = h^{1 / 2} {ln}^{β} (1 / h)$ , $β > 1 / 2$ .

Let ${(ξ_{n})}_{n \geq 1}$ be the polygonal partial sums processes built on the linear processes $X_{n} = \sum_{i \geq 0} a_{i} (ϵ_{n - i})$ , $n \geq 1$ , where ${(ϵ_{i})}_{i \in ℤ}$ are i.i.d., centered random elements in some separable Hilbert space $ℍ$ and the $a_{i}$ ’s are bounded linear operators $ℍ \to ℍ$ , with $\sum_{i \geq 0} ∥ a_{i} ∥ < \infty$ . We investigate functional central limit theorem for $ξ_{n}$ in the Hölder spaces $H_{ρ}^{o} (ℍ)$ of functions $x : [0, 1] \to ℍ$ such that $∥ x (t + h) - x (t) ∥ = o (ρ (h))$ uniformly in $t$ , where $ρ (h) = h^{α} L (1 / h)$ , $0 \leq h \leq 1$ with $0 < α \leq 1 / 2$ and $L$ slowly varying at infinity. We obtain the $H_{ρ}^{o} (ℍ)$ weak convergence of $ξ_{n}$ to some $ℍ$ valued brownian motion under the optimal assumption that for any $c > 0$ , $t P (∥ ϵ_{0} ∥ > c t^{1 / 2} ρ (1 / t)) = o (1)$ when $t$ tends to infinity, subject to some mild restriction on $L$ in the boundary case $α = 1 / 2$ . Our result holds in particular with the weight functions $ρ (h) = h^{1 / 2} {ln}^{β} (1 / h)$ , $β > 1 / 2$ .

DOI : 10.1051/ps:2008011

Classification : 60F17, 60B12
Mots-clés : central limit theorem in Banach spaces, Hölder space, functional central limit theorem, linear process, partial sums process

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     author = {Ra\v{c}kauskas, Alfredas and Suquet, Charles},
     title = {H\"olderian invariance principle for hilbertian linear processes},
     journal = {ESAIM: Probability and Statistics},
     pages = {261--275},
     publisher = {EDP-Sciences},
     volume = {13},
     year = {2009},
     doi = {10.1051/ps:2008011},
     mrnumber = {2528083},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ps:2008011/}
}

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Račkauskas, Alfredas; Suquet, Charles. Hölderian invariance principle for hilbertian linear processes. ESAIM: Probability and Statistics, Tome 13 (2009), pp. 261-275. doi : 10.1051/ps:2008011. https://www.numdam.org/articles/10.1051/ps:2008011/

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