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The sample ACF of a simple bilinear process

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  • Basrak, Bojan
  • Davis, Richard A.
  • Mikosch, Thomas

Abstract

We consider a simple bilinear process Xt=aXt-1+bXt-1Zt-1+Zt, where (Zt) is a sequence of iid N(0,1) random variables. It follows from a result by Kesten (1973, Acta Math. 131, 207-248) that Xt has a distribution with regularly varying tails of index [alpha]>0 provided the equation Ea+bZ1u=1 has the solution u=[alpha]. We study the limit behaviour of the sample autocorrelations and autocovariances of this heavy-tailed non-linear process. Of particular interest is the case when [alpha]

Suggested Citation

  • Basrak, Bojan & Davis, Richard A. & Mikosch, Thomas, 1999. "The sample ACF of a simple bilinear process," Stochastic Processes and their Applications, Elsevier, vol. 83(1), pages 1-14, September.
    • Handle: RePEc:eee:spapps:v:83:y:1999:i:1:p:1-14
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    References listed on IDEAS

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    1. Davis, Richard & Resnick, Sidney, 1985. "More limit theory for the sample correlation function of moving averages," Stochastic Processes and their Applications, Elsevier, vol. 20(2), pages 257-279, September.
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    Cited by:

    1. Aknouche, Abdelhakim & Guerbyenne, Hafida, 2009. "Periodic stationarity of random coefficient periodic autoregressions," Statistics & Probability Letters, Elsevier, vol. 79(7), pages 990-996, April.
    2. Pereira, I. & Scotto, M.G., 2006. "On the non-negative first-order exponential bilinear time series model," Statistics & Probability Letters, Elsevier, vol. 76(9), pages 931-938, May.
    3. Shiqing Ling & Liang Peng & Fukang Zhu, 2015. "Inference For A Special Bilinear Time-Series Model," Journal of Time Series Analysis, Wiley Blackwell, vol. 36(1), pages 61-66, January.
    4. Sakineh Ramezani & Mehrnaz Mohammadpour, 2022. "Integer-valued Bilinear Model with Dependent Counting Series," Methodology and Computing in Applied Probability, Springer, vol. 24(1), pages 321-343, March.
    5. Abdelhakim Aknouche & Nadia Rabehi, 2010. "On an independent and identically distributed mixture bilinear time‐series model," Journal of Time Series Analysis, Wiley Blackwell, vol. 31(2), pages 113-131, March.

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