Hölderian invariance principle for hilbertian linear processes
ESAIM: Probability and Statistics, Tome 13 (2009), pp. 261-275.

Soit (ξ n ) n1 le processus polygonal de sommes partielles bâti sur le processus linéaire X n = i0 a i (ϵ n-i ), n1, les (ϵ i ) i étant des éléments aléatoires i.i.d., centrés d’un espace de Hilbert séparable et les a i ’s des opérateurs linéaires bornés , vérifiant i0 a i <. Nous étudions le théorème limite central fonctionnel pour ξ n dans les espaces de Hölder H ρ o () de fonctions x:[0,1] vérifiant x(t+h)-x(t)=o(ρ(h)) uniformément en t, où ρ(h)=h α L(1/h), 0h1 avec 0<α1/2 et L à variation lente. Nous prouvons la convergence en loi dans H ρ o () de ξ n vers un mouvement brownien à valeurs dans , sous la condition optimale que pour tout c>0, tP(ϵ 0 >ct 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 ) n1 be the polygonal partial sums processes built on the linear processes X n = i0 a i (ϵ n-i ), n1, where (ϵ i ) i are i.i.d., centered random elements in some separable Hilbert space and the a i ’s are bounded linear operators , with i0 a i <. We investigate functional central limit theorem for ξ n in the Hölder spaces H ρ o () of functions x:[0,1] such that x(t+h)-x(t)=o(ρ(h)) uniformly in t, where ρ(h)=h α L(1/h), 0h1 with 0<α1/2 and L slowly varying at infinity. We obtain the H ρ o () weak convergence of ξ n to some valued brownian motion under the optimal assumption that for any c>0, tP(ϵ 0 >ct 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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     title = {H\"olderian invariance principle for hilbertian linear processes},
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     url = {http://archive.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. http://archive.numdam.org/articles/10.1051/ps:2008011/

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