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The asymptotic behavior of quadratic forms in heavy-tailed strongly dependent random variables

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  • Kokoszka, Piotr S.
  • Taqqu, Murad S.

Abstract

Suppose that Xt = [summation operator][infinity]j=0cjZt-j is a stationary linear sequence with regularly varying cj's and with innovations {Zj} that have infinite variance. Such a sequence can exhibit both high variability and strong dependence. The quadratic form 89 plays an important role in the estimation of the intensity of strong dependence. In contrast with the finite variance case, n-1/2(Qn - EQn) does not converge to a Gaussian distribution. We provide conditions on the cj's and on \gh for the quadratic form Qn, adequately normalized and randomly centered, to converge to a stable law of index [alpha], 1

Suggested Citation

  • Kokoszka, Piotr S. & Taqqu, Murad S., 1997. "The asymptotic behavior of quadratic forms in heavy-tailed strongly dependent random variables," Stochastic Processes and their Applications, Elsevier, vol. 66(1), pages 21-40, February.
    • Handle: RePEc:eee:spapps:v:66:y:1997:i:1:p:21-40
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    Cited by:

    1. Bhansali, R.J. & Giraitis, L. & Kokoszka, P.S., 2007. "Approximations and limit theory for quadratic forms of linear processes," Stochastic Processes and their Applications, Elsevier, vol. 117(1), pages 71-95, January.
    2. Chan, Ngai Hang & Zhang, Rong-Mao, 2013. "Limit theory of quadratic forms of long-memory linear processes with heavy-tailed GARCH innovations," Journal of Multivariate Analysis, Elsevier, vol. 120(C), pages 18-33.
    3. van Delft, Anne, 2020. "A note on quadratic forms of stationary functional time series under mild conditions," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 4206-4251.

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    Keywords

    60F05 60E07;

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