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Functional Convergence of Linear Processes with Heavy-Tailed Innovations

Author

Listed:
  • Raluca Balan

    (University of Ottawa)

  • Adam Jakubowski

    (Nicolaus Copernicus University)

  • Sana Louhichi

    (Institut de mathématiques appliquées de Grenoble)

Abstract

We study convergence in law of partial sums of linear processes with heavy-tailed innovations. In the case of summable coefficients, necessary and sufficient conditions for the finite dimensional convergence to an $$\alpha $$ α -stable Lévy Motion are given. The conditions lead to new, tractable sufficient conditions in the case $$\alpha \le 1$$ α ≤ 1 . In the functional setting, we complement the existing results on $$M_1$$ M 1 -convergence, obtained for linear processes with nonnegative coefficients by Avram and Taqqu (Ann Probab 20:483–503, 1992) and improved by Louhichi and Rio (Electr J Probab 16(89), 2011), by proving that in the general setting partial sums of linear processes are convergent on the Skorokhod space equipped with the $$S$$ S topology, introduced by Jakubowski (Electr J Probab 2(4), 1997).

Suggested Citation

  • Raluca Balan & Adam Jakubowski & Sana Louhichi, 2016. "Functional Convergence of Linear Processes with Heavy-Tailed Innovations," Journal of Theoretical Probability, Springer, vol. 29(2), pages 491-526, June.
    • Handle: RePEc:spr:jotpro:v:29:y:2016:i:2:d:10.1007_s10959-014-0581-9
    DOI: 10.1007/s10959-014-0581-9
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. Liu, Hui & Xiong, Yudan & Xu, Fangjun, 2024. "Limit theorems for functionals of long memory linear processes with infinite variance," Stochastic Processes and their Applications, Elsevier, vol. 167(C).
    2. Peligrad, Magda & Sang, Hailin & Xiao, Yimin & Yang, Guangyu, 2022. "Limit theorems for linear random fields with innovations in the domain of attraction of a stable law," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 596-621.

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    2. Peligrad, Magda & Sang, Hailin & Xiao, Yimin & Yang, Guangyu, 2022. "Limit theorems for linear random fields with innovations in the domain of attraction of a stable law," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 596-621.

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