IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v152y2022icp208-232.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414922001569
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2022.06.021?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. 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.
    2. 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.
    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. Liu, Ming, 2000. "Modeling long memory in stock market volatility," Journal of Econometrics, Elsevier, vol. 99(1), pages 139-171, November.
    5. 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.
    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, November.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    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. 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.
    3. Steland, Ansgar & von Sachs, Rainer, 2016. "Asymptotics for High–Dimensional Covariance Matrices and Quadratic Forms with Applications to the Trace Functional and Shrinkage," LIDAM Discussion Papers ISBA 2016038, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Ehsan Azmoodeh & Yuliya Mishura & Farzad Sabzikar, 2022. "How Does Tempering Affect the Local and Global Properties of Fractional Brownian Motion?," Journal of Theoretical Probability, Springer, vol. 35(1), pages 484-527, March.
    5. Abry, Patrice & Didier, Gustavo, 2018. "Wavelet eigenvalue regression for n-variate operator fractional Brownian motion," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 75-104.
    6. Steland, Ansgar & von Sachs, Rainer, 2018. "Asymptotics for high-dimensional covariance matrices and quadratic forms with applications to the trace functional and shrinkage," Stochastic Processes and their Applications, Elsevier, vol. 128(8), pages 2816-2855.
    7. 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.
    8. Araya, Héctor & Tudor, Ciprian A., 2019. "Behavior of the Hermite sheet with respect to theHurst index," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2582-2605.
    9. Patrice Abry & Gustavo Didier & Hui Li, 2019. "Two-step wavelet-based estimation for Gaussian mixed fractional processes," Statistical Inference for Stochastic Processes, Springer, vol. 22(2), pages 157-185, July.
    10. 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.
    11. Kechagias, Stefanos & Pipiras, Vladas & Zoubouloglou, Pavlos, 2024. "Cyclical long memory: Decoupling, modulation, and modeling," Stochastic Processes and their Applications, Elsevier, vol. 175(C).
    12. Paul Doukhan & Olivier Wintenberger, 2005. "An Invariance Principle for New Weakly Dependent Stationary Models using Sharp Moment Assumptions," Working Papers 2005-51, Center for Research in Economics and Statistics.
    13. Shi, Yanlin & Ho, Kin-Yip, 2015. "Modeling high-frequency volatility with three-state FIGARCH models," Economic Modelling, Elsevier, vol. 51(C), pages 473-483.
    14. Kerstin Gärtner & Mark Podolskij, 2014. "On non-standard limits of Brownian semi-stationary," CREATES Research Papers 2014-50, Department of Economics and Business Economics, Aarhus University.
    15. Nourdin, Ivan & Peccati, Giovanni & Podolskij, Mark, 2011. "Quantitative Breuer-Major theorems," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 793-812, April.
    16. Shao, Xiaofeng, 2011. "A bootstrap-assisted spectral test of white noise under unknown dependence," Journal of Econometrics, Elsevier, vol. 162(2), pages 213-224, June.
    17. Debashis Mondal & Donald Percival, 2010. "Wavelet variance analysis for gappy time series," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(5), pages 943-966, October.
    18. Smith, Aaron, 2005. "Level Shifts and the Illusion of Long Memory in Economic Time Series," Journal of Business & Economic Statistics, American Statistical Association, vol. 23, pages 321-335, July.
    19. Fabio Vanni & David Lambert, 2023. "A detection analysis for temporal memory patterns at different time-scales," Papers 2309.12034, arXiv.org.
    20. Surgailis, Donatas & Teyssière, Gilles & Vaiciulis, Marijus, 2008. "The increment ratio statistic," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 510-541, March.

    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:eee:spapps:v:152:y:2022:i:c:p:208-232. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    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.