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 0ρ (B) be the Hölder space of functions x:[0,1]↦B, such that ∥x(t+h)−x(t)∥=o(ρ(h)), uniformly in t. We characterize the weak convergence in H 0ρ (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.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
references
Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1987). Regular variation. In Encyclopaedia of Mathematics and Its Applications, Cambridge University Press.
Ciesielski, Z. (1960). On the isomorphisms of the spaces Hα and m. Bull. Acad. Pol. Sci. Ser. Sci. Math. Phys. 8, 217-222.
Erickson, R. V. (1981). Lipschitz smoothness and convergence with applications to the central limit theorem for summation processes. Ann. Probab. 9, 831-851.
Hamadouche, D. (2000). Invariance principles in Hölder spaces. Portugal. Math. 57, 127-151.
Kerkyacharian, G., and Roynette, B. (1991). FrUne démonstration simple des théorèmes de Kolmogorov, Donsker, et Ito–Nisio. C. R. Acad. Sci. Paris Sér. I Math. 312, 877-882.
Kuelbs, J. (1973). The invariance principle for Banach space valued random variables. J. Multivariate Anal. 3, 161-172.
Lamperti, J. (1962). On convergence of stochastic processes. Trans. Amer. Math. Soc. 104, 430-435.
Ledoux, M., and Talagrand, M. (1991). Probability in Banach Spaces, Springer-Verlag, Berlin, Heidelberg.
Lévy, P. (1937). Théorie de l'addition des variables aléatoires, 2nd ed. 1954, Gauthier-Villars, Paris.
Račkauskas, A., and Suquet, C. (1998). On the Hölderian functional central limit theorem for i.i.d. random elements in Banach space. In Berkes, I., Csáki, E., Csörgö, M. (eds.), Limit Theorems in Probability and Statistics, Balatonlelle 1999, János Bolyai Mathematical Society, Budapest, 2002, Vol.2, pp. 485-498.
Račkauskas, A., and Suquet, C. (2001). Invariance principles for adaptive self-normalized partial sums processes. Stochastic Process. Appl. 95, 63-81.
Račkauskas, A., and Suquet, C. (2001). Hölder versions of Banach spaces valued random fields. Georgian Math. J. 8(2), 347-362.
Semadeni, Z. (1982). Schauder bases in Banach spaces of continuous functions. In Lecture Notes in Math., Vol.918, Springer-Verlag, Berlin, Heidelberg, New York.
Suquet, C. (1999). Tightness in Schauder decomposable Banach spaces. Amer. Math. Soc. Transl. Ser. 2 193, 201-224.
Talagrand, M. (1989). Isoperimetry and integrability of the sum of independent Banach-space valued random variables. Ann. Probab. 17, 1546-1570.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Račkauskas, A., Suquet, C. Necessary and Sufficient Condition for the Functional Central Limit Theorem in Hölder Spaces. Journal of Theoretical Probability 17, 221–243 (2004). https://doi.org/10.1023/B:JOTP.0000020482.66224.6c
Issue Date:
DOI: https://doi.org/10.1023/B:JOTP.0000020482.66224.6c