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On almost sure limit theorems for heavy-tailed products of long-range dependent linear processes

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  • Kouritzin, Michael A.
  • Paul, Sounak

Abstract

Marcinkiewicz strong law of large numbers, n−1p∑k=1n(dk−d)→0 almost surely with p∈(1,2), are developed for products dk=∏r=1sxk(r), where xk(r)=∑l=−∞∞ck−l(r)ξl(r) are two-sided linear processes with coefficients {cl(r)}l∈Z and i.i.d. zero-mean innovations {ξl(r)}l∈Z. The decay of the coefficients cl(r) as |l|→∞, can be slow enough for {xk(r)} to have long memory while {dk} can have heavy tails. The long-range dependence and heavy tails for {dk} are handled simultaneously and a decoupling property shows the convergence rate is dictated by the worst of long-range dependence and heavy tails, but not their combination. The Marcinkiewicz strong law of large numbers is also extended to the multivariate linear process case.

Suggested Citation

  • Kouritzin, Michael A. & Paul, Sounak, 2022. "On almost sure limit theorems for heavy-tailed products of long-range dependent linear processes," Stochastic Processes and their Applications, Elsevier, vol. 152(C), pages 208-232.
    • Handle: RePEc:eee:spapps:v:152:y:2022:i:c:p:208-232
    DOI: 10.1016/j.spa.2022.06.021
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    References listed on IDEAS

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    1. Kouritzin, Michael A., 1995. "Strong approximation for cross-covariances of linear variables with long-range dependence," Stochastic Processes and their Applications, Elsevier, vol. 60(2), pages 343-353, December.
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    3. Breuer, Péter & Major, Péter, 1983. "Central limit theorems for non-linear functionals of Gaussian fields," Journal of Multivariate Analysis, Elsevier, vol. 13(3), pages 425-441, September.
    4. Sana Louhichi & Philippe Soulier, 2000. "Marcinkiewicz–Zygmund Strong Laws for Infinite Variance Time Series," Statistical Inference for Stochastic Processes, Springer, vol. 3(1), pages 31-40, January.
    5. Biao Wu, Wei & Min, Wanli, 2005. "On linear processes with dependent innovations," Stochastic Processes and their Applications, Elsevier, vol. 115(6), pages 939-958, June.
    6. Bai, Shuyang & Taqqu, Murad S., 2015. "Convergence of long-memory discrete kth order Volterra processes," Stochastic Processes and their Applications, Elsevier, vol. 125(5), pages 2026-2053.
    7. Pipiras,Vladas & Taqqu,Murad S., 2017. "Long-Range Dependence and Self-Similarity," Cambridge Books, Cambridge University Press, number 9781107039469.
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    Cited by:

    1. Yu Zhang, 2023. "Asymptotic Normality of M-Estimator in Linear Regression Model with Asymptotically Almost Negatively Associated Errors," Mathematics, MDPI, vol. 11(18), pages 1-16, September.

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