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Central Limit Theorems for Weighted Sums of Dependent Random Vectors in Hilbert Spaces via the Theory of the Regular Variation

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  • Ta Cong Son

    (VNU University of Science, Vietnam National University)

  • Le Van Dung

    (The University of Da Nang - University of Science and Education)

Abstract

In this paper, based on the theory of regularly varying functions we study central limit theorems for the weighted sum $$S_n=\sum _{j=1}^{m_n}c_{nj}X_{nj}$$ S n = ∑ j = 1 m n c nj X nj , where $$(X_{nj};1\le j \le m_n,n\ge 1)$$ ( X nj ; 1 ≤ j ≤ m n , n ≥ 1 ) is a Hilbert-space-valued identically distributed martingale difference array and $$(c_{nj};1\le j \le m_n,n\ge 1)$$ ( c nj ; 1 ≤ j ≤ m n , n ≥ 1 ) is an array of real numbers. As an application, we present a central limit theorem for moving average processes of martingale differences.

Suggested Citation

  • Ta Cong Son & Le Van Dung, 2022. "Central Limit Theorems for Weighted Sums of Dependent Random Vectors in Hilbert Spaces via the Theory of the Regular Variation," Journal of Theoretical Probability, Springer, vol. 35(2), pages 988-1012, June.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:2:d:10.1007_s10959-021-01079-4
    DOI: 10.1007/s10959-021-01079-4
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    References listed on IDEAS

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    1. Crimaldi, Irene & Pratelli, Luca, 2005. "Convergence results for multivariate martingales," Stochastic Processes and their Applications, Elsevier, vol. 115(4), pages 571-577, April.
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    3. Pipiras,Vladas & Taqqu,Murad S., 2017. "Long-Range Dependence and Self-Similarity," Cambridge Books, Cambridge University Press, number 9781107039469.
    4. Dehling, Herold & Sharipov, Olimjon Sh. & Wendler, Martin, 2015. "Bootstrap for dependent Hilbert space-valued random variables with application to von Mises statistics," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 200-215.
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