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Necessary and Sufficient Condition for the Functional Central Limit Theorem in Hölder Spaces

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  • Alfredas Račkauskas

    (Vilnius University)

  • Charles Suquet

    (Université Lille I)

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 t)=o(t −p(α)) where p(α)=1/(1/2−α). This completes Lamperti (1962) invariance principle.

Suggested Citation

  • Alfredas Račkauskas & Charles Suquet, 2004. "Necessary and Sufficient Condition for the Functional Central Limit Theorem in Hölder Spaces," Journal of Theoretical Probability, Springer, vol. 17(1), pages 221-243, January.
    • Handle: RePEc:spr:jotpro:v:17:y:2004:i:1:d:10.1023_b:jotp.0000020482.66224.6c
    DOI: 10.1023/B:JOTP.0000020482.66224.6c
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    References listed on IDEAS

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    1. Kuelbs, J., 1973. "The invariance principle for Banach space valued random variables," Journal of Multivariate Analysis, Elsevier, vol. 3(2), pages 161-172, June.
    2. Rackauskas, Alfredas & Suquet, Charles, 2001. "Invariance principles for adaptive self-normalized partial sums processes," Stochastic Processes and their Applications, Elsevier, vol. 95(1), pages 63-81, September.
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    Cited by:

    1. Davide Giraudo, 2017. "Holderian Weak Invariance Principle for Stationary Mixing Sequences," Journal of Theoretical Probability, Springer, vol. 30(1), pages 196-211, March.
    2. Horvath, Blanka & Jacquier, Antoine & Muguruza, Aitor & Søjmark, Andreas, 2024. "Functional central limit theorems for rough volatility," LSE Research Online Documents on Economics 122848, London School of Economics and Political Science, LSE Library.
    3. Blanka Horvath & Antoine Jacquier & Aitor Muguruza & Andreas Søjmark, 2024. "Functional central limit theorems for rough volatility," Finance and Stochastics, Springer, vol. 28(3), pages 615-661, July.

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