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

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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


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Necessary and sufficient condition for the functional centrallimittheoreminH¨olderspaces ByAlfredasRacˇkauskas 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 oρ ( 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: Alfredas.Rackauskas@maf.vu.lt 2 CNRSFRE2222,LaboratoiredeMath´ematiquesApplique´es,Bˆat.M2, Universit´eLilleI,F-59655VilleneuvedAscqCedex,France. E-mail: Charles.Suquet@univ-lille1.fr 3 Research supported by a cooperation agreement CNRS/LITHUANIA (4714). 4 Partially supported by Vilnius Institute of Mathematics and Informatics. 1
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