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://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. http://www.numdam.org/articles/10.1051/ps:2008011/

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

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

[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). | MR | Zbl

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

[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. | MR | Zbl

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

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

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

[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. | MR | Zbl

[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. | MR | Zbl

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

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

[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. | MR | Zbl

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

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

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