IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v35y2022i2d10.1007_s10959-021-01079-4.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-021-01079-4
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10959-021-01079-4?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Crimaldi, Irene & Pratelli, Luca, 2005. "Convergence results for multivariate martingales," Stochastic Processes and their Applications, Elsevier, vol. 115(4), pages 571-577, April.
    2. Magda Peligrad & Hailin Sang, 2013. "Central Limit Theorem for Linear Processes with Infinite Variance," Journal of Theoretical Probability, Springer, vol. 26(1), pages 222-239, March.
    3. 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.
    4. Pipiras,Vladas & Taqqu,Murad S., 2017. "Long-Range Dependence and Self-Similarity," Cambridge Books, Cambridge University Press, number 9781107039469.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Bai, Shuyang & Taqqu, Murad S., 2019. "Sensitivity of the Hermite rank," Stochastic Processes and their Applications, Elsevier, vol. 129(3), pages 822-840.
    2. Shuyang Bai, 2022. "Limit Theorems for Conservative Flows on Multiple Stochastic Integrals," Journal of Theoretical Probability, Springer, vol. 35(2), pages 917-948, June.
    3. Yuanhua Feng & Wolfgang Karl Härdle, 2021. "Uni- and multivariate extensions of the sinh-arcsinh normal distribution applied to distributional regression," Working Papers CIE 142, Paderborn University, CIE Center for International Economics.
    4. Ran Wang & Yimin Xiao, 2022. "Exact Uniform Modulus of Continuity and Chung’s LIL for the Generalized Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2442-2479, December.
    5. Berkes, István & Horváth, Lajos & Rice, Gregory, 2016. "On the asymptotic normality of kernel estimators of the long run covariance of functional time series," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 150-175.
    6. Grahovac, Danijel & Leonenko, Nikolai N. & Taqqu, Murad S., 2018. "Intermittency of trawl processes," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 235-242.
    7. Nourdin, Ivan & Nualart, David & Peccati, Giovanni, 2021. "The Breuer–Major theorem in total variation: Improved rates under minimal regularity," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 1-20.
    8. Battey, H.S. & Cox, D.R., 2022. "Some aspects of non-standard multivariate analysis," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    9. María E. Sousa-Vieira & Manuel Fernández-Veiga, 2023. "Study of Coded ALOHA with Multi-User Detection under Heavy-Tailed and Correlated Arrivals," Future Internet, MDPI, vol. 15(4), pages 1-18, March.
    10. Patrice Abry & Yannick Malevergne & Herwig Wendt & Marc Senneret & Laurent Jaffrès & Blaise Liaustrat, 2019. "Shuffling for understanding multifractality, application to asset price time series," Post-Print hal-02361738, HAL.
    11. Kubiv Stepan, 2019. "Approximations and forecasting quasi-stationary processes with sudden runs," Technology audit and production reserves, 4(48) 2019, Socionet;Technology audit and production reserves, vol. 4(4(48)), pages 37-39.
    12. Rudy Morel & Gaspar Rochette & Roberto Leonarduzzi & Jean-Philippe Bouchaud & St'ephane Mallat, 2022. "Scale Dependencies and Self-Similar Models with Wavelet Scattering Spectra," Papers 2204.10177, arXiv.org, revised Jun 2023.
    13. Grahovac, Danijel, 2022. "Intermittency in the small-time behavior of Lévy processes," Statistics & Probability Letters, Elsevier, vol. 187(C).
    14. Baek, Changryong & Gates, Katheleen M. & Leinwand, Benjamin & Pipiras, Vladas, 2021. "Two sample tests for high-dimensional autocovariances," Computational Statistics & Data Analysis, Elsevier, vol. 153(C).
    15. Johann Gehringer & Xue-Mei Li, 2022. "Functional Limit Theorems for the Fractional Ornstein–Uhlenbeck Process," Journal of Theoretical Probability, Springer, vol. 35(1), pages 426-456, March.
    16. Didier, Gustavo & Meerschaert, Mark M. & Pipiras, Vladas, 2018. "Domain and range symmetries of operator fractional Brownian fields," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 39-78.
    17. Obayda Assaad & Ciprian A. Tudor, 2020. "Parameter identification for the Hermite Ornstein–Uhlenbeck process," Statistical Inference for Stochastic Processes, Springer, vol. 23(2), pages 251-270, July.
    18. Georgia Papadogeorgou & Kosuke Imai & Jason Lyall & Fan Li, 2022. "Causal inference with spatio‐temporal data: Estimating the effects of airstrikes on insurgent violence in Iraq," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(5), pages 1969-1999, November.
    19. Hien, N.T.T. & Thanh, L.V., 2015. "On the weak laws of large numbers for sums of negatively associated random vectors in Hilbert spaces," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 236-245.
    20. Yuanhua Feng & Jan Beran & Sebastian Letmathe & Sucharita Ghosh, 2020. "Fractionally integrated Log-GARCH with application to value at risk and expected shortfall," Working Papers CIE 137, Paderborn University, CIE Center for International Economics.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:35:y:2022:i:2:d:10.1007_s10959-021-01079-4. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.