Hölderian invariance principle for hilbertian linear processes

Račkauskas, Alfredas; Suquet, Charles

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

Soit ${\left({\xi }_n\right)}_{n\ge 1}$ le processus polygonal de sommes partielles bâti sur le processus linéaire $X_n={\sum }_{i\ge 0}{a}_i\left({ϵ}_{n-i}\right)$, $n\ge 1$, les ${\left({ϵ}_i\right)}_{i\in \mathbb{Z}}$ 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 ${\sum }_{i\ge 0}\parallel a_i\parallel <\infty$. Nous étudions le théorème limite central fonctionnel pour ${\xi }_n$ dans les espaces de Hölder ${\mathrm{H}}_{\rho }^o\left(ℍ\right)$ de fonctions $x:[0,1]\to ℍ$ vérifiant $\parallel x(t+h)-x\left(t\right)\parallel =o\left(\rho \right(h\left)\right)$ uniformément en $t$, où $\rho \left(h\right)=h^{\alpha }L(1/h)$, $0\le h\le 1$ avec $0<\alpha \le 1/2$ et $L$ à variation lente. Nous prouvons la convergence en loi dans ${\mathrm{H}}_{\rho }^o\left(ℍ\right)$ de ${\xi }_n$ vers un mouvement brownien à valeurs dans $ℍ$, sous la condition optimale que pour tout $c>0$, $tP(\parallel {ϵ}_0\parallel >c{t}^{1/2}\rho (1/t))=o\left(1\right)$ quand $t$ tend vers l’infini, au prix dans le cas limite $\alpha =1/2$ d’une légère restriction sur $L$. Notre résultat s’applique en particulier au cas $\rho \left(h\right)=h^{1/2}{ln}^{\beta }(1/h)$, $\beta >1/2$.

Let ${\left({\xi }_n\right)}_{n\ge 1}$ be the polygonal partial sums processes built on the linear processes $X_n={\sum }_{i\ge 0}{a}_i\left({ϵ}_{n-i}\right)$, $n\ge 1$, where ${\left({ϵ}_i\right)}_{i\in \mathbb{Z}}$ 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 ${\sum }_{i\ge 0}\parallel a_i\parallel <\infty$. We investigate functional central limit theorem for ${\xi }_n$ in the Hölder spaces ${\mathrm{H}}_{\rho }^o\left(ℍ\right)$ of functions $x:[0,1]\to ℍ$ such that $\parallel x(t+h)-x\left(t\right)\parallel =o\left(\rho \right(h\left)\right)$ uniformly in $t$, where $\rho \left(h\right)=h^{\alpha }L(1/h)$, $0\le h\le 1$ with $0<\alpha \le 1/2$ and $L$ slowly varying at infinity. We obtain the ${\mathrm{H}}_{\rho }^o\left(ℍ\right)$ weak convergence of ${\xi }_n$ to some $ℍ$ valued brownian motion under the optimal assumption that for any $c>0$, $tP(\parallel {ϵ}_0\parallel >c{t}^{1/2}\rho (1/t))=o\left(1\right)$ when $t$ tends to infinity, subject to some mild restriction on $L$ in the boundary case $\alpha =1/2$. Our result holds in particular with the weight functions $\rho \left(h\right)=h^{1/2}{ln}^{\beta }(1/h)$, $\beta >1/2$.

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