Necessary and sufficient condition for the functional central limit theorem in Holder spaces

Charles Suquet2

Description

Necessary and sufficient condition for the functional central limit theorem in Holder spaces By Alfredas Racˇkauskas1,3,4 and Charles Suquet2,3 Revised version September 10, 2003 Abstract Let (Xi)i≥1 be an i.i.d. sequence of random elements in the Banach space B, Sn := X1+· · ·+Xn and ?n be the random polygonal line with vertices (k/n, Sk), k = 0, 1, . . . , n. Put ?(h) = h?L(1/h), 0 ≤ h ≤ 1 with 0 < ? ≤ 1/2 and L slowly varying at infinity. Let Ho?(B) be the Holder space of functions x : [0, 1] 7? B, such that ||x(t+ h)? x(t)|| = o(?(h)), uniformly in t. We characterize the weak convergence in Ho?(B) of n?1/2?n to a Brownian motion. In the special case where B = R and ?(h) = h?, our necessary and sufficient conditions for such convergence are EX1 = 0 and P(|X1| > t) = o(t?p(?)) where p(?) = 1/(1/2 ? ?).

• general ?


• self-normalized partial


• holder spaces


• sums processes


• banach space


• valued coefficients


• invariance principle


• x1


• then ? fulfills

Sujets

• Vilnius University
• Suquet
• Banach space
• Invariance principle

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 24
• Langue: English

Extrait

Necessary and sufficient condition for the functional centrallimittheoreminH¨olderspaces ByAlfredasRac ˇ kauskas 1 , 3 , 4 and Charles Suquet 2 , 3 Revised version September 10, 2003

Abstract Let ( X i ) i ≥ 1 be an i.i.d. sequence of random elements in the Banach space B , S n := X 1 + ∙ ∙ ∙ + X n and ξ n be the random polygonal line with vertices ( k/n, S k ), k = 0 , 1 , . . . , n . Put ρ ( h ) = h α L (1 /h ), 0 ≤ h ≤ 1 with 0 < α ≤ 1 / 2 and L slowly varying at infinity. Let H o ρ ( B ) be the H¨olderspaceoffunctions x : [0 , 1] 7→ B , such that || x ( t + h ) − x ( t ) || = o ( ρ ( h )), uniformly in t . We characterize the weak convergence in H ρo ( B ) of n − 1 / 2 ξ n to a Brownian motion. In the special case where B = R and ρ ( h ) = h α , our necessary and sufficient conditions for such convergence are E X 1 = 0 and P ( | X 1 | > t ) = o ( t − p ( α ) ) where p ( α ) = 1 / (1 / 2 − α ). This completes Lamperti (1962) invariance principle. MSC 2000 subject classifications . Primary-60F17; secondary-60B12. Key words and phrases .CentrallimittheoreminBanachspaces,Ho¨lder space, invariance principle, partial sums process.

1 Department of Mathematics, Vilnius University, Naugarduko 24, Lt-2006 Vilnius, Lithuania. E-mail: [e-mail] 2 CNRSFRE2222,LaboratoiredeMath ´ ematiquesApplique ´ es,B ˆ at.M2, Universit ´ eLilleI,F-59655Villeneuved’AscqCedex,France. E-mail: [e-mail] 3 Research supported by a cooperation agreement CNRS/LITHUANIA (4714). 4 Partially supported by Vilnius Institute of Mathematics and Informatics. 1