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On the Central Limit Theorem and Its Weak Invariance Principle for Strongly Mixing Sequences with Values in a Hilbert Space via Martingale Approximation

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  • Florence Merlevède

    (Université Paris VI)

Abstract

In this paper we not only prove an extension to Hilbert spaces of a sharp central limit theorem for strongly real-valued mixing sequences, but also slightly improve it. The proof is mainly based on the Bernstein blocking technique and approximations by martingale differences. Moreover, we derive also the corresponding functional central limit theorem.

Suggested Citation

  • Florence Merlevède, 2003. "On the Central Limit Theorem and Its Weak Invariance Principle for Strongly Mixing Sequences with Values in a Hilbert Space via Martingale Approximation," Journal of Theoretical Probability, Springer, vol. 16(3), pages 625-653, July.
    • Handle: RePEc:spr:jotpro:v:16:y:2003:i:3:d:10.1023_a:1025668415566
    DOI: 10.1023/A:1025668415566
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    References listed on IDEAS

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    1. Florence Merlevède & Magda Peligrad & Sergey Utev, 1997. "Sharp Conditions for the CLT of Linear Processes in a Hilbert Space," Journal of Theoretical Probability, Springer, vol. 10(3), pages 681-693, July.
    2. Kuelbs, J., 1973. "The invariance principle for Banach space valued random variables," Journal of Multivariate Analysis, Elsevier, vol. 3(2), pages 161-172, June.
    3. Chen, Xiaohong & White, Halbert, 1998. "Central Limit And Functional Central Limit Theorems For Hilbert-Valued Dependent Heterogeneous Arrays With Applications," Econometric Theory, Cambridge University Press, vol. 14(2), pages 260-284, April.
    4. Richard C. Bradley, 1997. "On Quantiles and the Central Limit Question for Strongly Mixing Sequences," Journal of Theoretical Probability, Springer, vol. 10(2), pages 507-555, April.
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