Hölderian invariance principle for hilbertian linear processes

Račkauskas, Alfredas; Suquet, Charles

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

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 $\rho \left(h\right)=h^{1/2}{ln}^{\beta }(1/h)$, $\beta >1/2$.

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 $\rho \left(h\right)=h^{1/2}{ln}^{\beta }(1/h)$, $\beta >1/2$.

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

[1] N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular variation. Encyclopaedia of Mathematics and its Applications. Cambridge University Press (1987). | Zbl | MR

[2] J. Dedecker and F. Merlevède, The conditional central limit theorem in Hilbert spaces. Stoch. Process. Appl. 108 (2003) 229-262. | Zbl | MR

[3] J. Dedecker, P. Doukhan, G. Lang, J.R. Leon, S. Louhichi and C. Prieur, Weak Dependence: With Examples and Applications, volume 190 of Lect. Notes Statist. Springer (2007). | Zbl | MR

[4] D. Hamadouche, Invariance principles in Hölder spaces. Portugal. Math. 57 (2000) 127-151. | Zbl | MR

[5] M. Juodis, A. Račkauskas and Ch. Suquet, Hölderian functional central limit theorems for linear processes. ALEA Lat. Am. J. Probab. Math. Stat. 5 (2009) 47-64. | Zbl | MR

[6] J. Kuelbs, The invariance principle for Banach space valued random variables. J. Multiv. Anal. 3 (1973) 161-172. | Zbl | MR

[7] J. Lamperti, On convergence of stochastic processes. Trans. Amer. Math. Soc. 104 (1962) 430-435. | Zbl | MR

[8] M. Ledoux and M. Talagrand, Probability in Banach Spaces. Springer-Verlag, Berlin, Heidelberg (1991). | Zbl | MR

[9] F. Merlevède, M. Peligrad and S. Utev, Sharp conditions for the CLT of linear processes in a Hilbert space. J. Theoret. Probab. 10 (1997) 681-693. | Zbl | MR

[10] F. Merlevède, M. Peligrad and S. Utev, Recent advances in invariance principles for stationary sequences. Probab. Surveys 3 (2006) 1-36. | MR

[11] A. Račkauskas and Ch. Suquet, Hölder versions of Banach spaces valued random fields. Georgian Math. J. 8 (2001) 347-362. | Zbl | MR

[12] A. Račkauskas and Ch. Suquet, Necessary and sufficient condition for the Hölderian functional central limit theorem. J. Theoret. Probab. 17 (2004) 221-243. | Zbl | MR

[13] A. Račkauskas and Ch. Suquet, Hölder norm test statistics for epidemic change. J. Statist. Plann. Inference 126 (2004) 495-520. | Zbl | MR

[14] A. Račkauskas and Ch. Suquet, Central limit theorems in Hölder topologies for Banach space valued random fields. Theor. Probab. Appl. 49 (2004) 109-125. | Zbl | MR

[15] A. Račkauskas and Ch. Suquet, Testing epidemic changes of infinite dimensional parameters. Stat. Inference Stoch. Process. 9 (2006) 111-134. | Zbl | MR

[16] M. Talagrand, Isoperimetry and integrability of the sum of independent Banach-space valued random variables. Ann. Probab. 17 (1989) 1546-1570. | Zbl | MR