Soit le processus polygonal de sommes partielles bâti sur le processus linéaire , , les étant des éléments aléatoires i.i.d., centrés d’un espace de Hilbert séparable et les ’s des opérateurs linéaires bornés , vérifiant . Nous étudions le théorème limite central fonctionnel pour dans les espaces de Hölder de fonctions vérifiant uniformément en , où , avec et à variation lente. Nous prouvons la convergence en loi dans de vers un mouvement brownien à valeurs dans , sous la condition optimale que pour tout , quand tend vers l’infini, au prix dans le cas limite d’une légère restriction sur . Notre résultat s’applique en particulier au cas , .
Let be the polygonal partial sums processes built on the linear processes , , where are i.i.d., centered random elements in some separable Hilbert space and the ’s are bounded linear operators , with . We investigate functional central limit theorem for in the Hölder spaces of functions such that uniformly in , where , with and slowly varying at infinity. We obtain the weak convergence of to some valued brownian motion under the optimal assumption that for any , when tends to infinity, subject to some mild restriction on in the boundary case . Our result holds in particular with the weight functions , .
Mots-clés : central limit theorem in Banach spaces, Hölder space, functional central limit theorem, linear process, partial sums process
@article{PS_2009__13__261_0, 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 = {http://archive.numdam.org/articles/10.1051/ps:2008011/} }
TY - JOUR AU - Račkauskas, Alfredas AU - Suquet, Charles TI - Hölderian invariance principle for hilbertian linear processes JO - ESAIM: Probability and Statistics PY - 2009 SP - 261 EP - 275 VL - 13 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps:2008011/ DO - 10.1051/ps:2008011 LA - en ID - PS_2009__13__261_0 ER -
%0 Journal Article %A Račkauskas, Alfredas %A Suquet, Charles %T Hölderian invariance principle for hilbertian linear processes %J ESAIM: Probability and Statistics %D 2009 %P 261-275 %V 13 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps:2008011/ %R 10.1051/ps:2008011 %G en %F PS_2009__13__261_0
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/
[1] Regular variation. Encyclopaedia of Mathematics and its Applications. Cambridge University Press (1987). | MR | Zbl
, and ,[2] The conditional central limit theorem in Hilbert spaces. Stoch. Process. Appl. 108 (2003) 229-262. | MR | Zbl
and ,[3] Weak Dependence: With Examples and Applications, volume 190 of Lect. Notes Statist. Springer (2007). | MR | Zbl
, , , , and ,[4] Invariance principles in Hölder spaces. Portugal. Math. 57 (2000) 127-151. | MR | Zbl
,[5] Hölderian functional central limit theorems for linear processes. ALEA Lat. Am. J. Probab. Math. Stat. 5 (2009) 47-64. | MR | Zbl
, and ,[6] The invariance principle for Banach space valued random variables. J. Multiv. Anal. 3 (1973) 161-172. | MR | Zbl
,[7] On convergence of stochastic processes. Trans. Amer. Math. Soc. 104 (1962) 430-435. | MR | Zbl
,[8] Probability in Banach Spaces. Springer-Verlag, Berlin, Heidelberg (1991). | MR | Zbl
and ,[9] Sharp conditions for the CLT of linear processes in a Hilbert space. J. Theoret. Probab. 10 (1997) 681-693. | MR | Zbl
, and ,[10] Recent advances in invariance principles for stationary sequences. Probab. Surveys 3 (2006) 1-36. | MR
, and ,[11] Hölder versions of Banach spaces valued random fields. Georgian Math. J. 8 (2001) 347-362. | MR | Zbl
and ,[12] Necessary and sufficient condition for the Hölderian functional central limit theorem. J. Theoret. Probab. 17 (2004) 221-243. | MR | Zbl
and ,[13] Hölder norm test statistics for epidemic change. J. Statist. Plann. Inference 126 (2004) 495-520. | MR | Zbl
and ,[14] Central limit theorems in Hölder topologies for Banach space valued random fields. Theor. Probab. Appl. 49 (2004) 109-125. | MR | Zbl
and ,[15] Testing epidemic changes of infinite dimensional parameters. Stat. Inference Stoch. Process. 9 (2006) 111-134. | MR | Zbl
and ,[16] Isoperimetry and integrability of the sum of independent Banach-space valued random variables. Ann. Probab. 17 (1989) 1546-1570. | MR | Zbl
,Cité par Sources :