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Approximations and limit theory for quadratic forms of linear processes

Author

Listed:
  • Bhansali, R.J.
  • Giraitis, L.
  • Kokoszka, P.S.

Abstract

The paper develops a limit theory for the quadratic form Qn,X in linear random variables X1,...,Xn which can be used to derive the asymptotic normality of various semiparametric, kernel, window and other estimators converging at a rate which is not necessarily n1/2. The theory covers practically all forms of linear serial dependence including long, short and negative memory, and provides conditions which can be readily verified thus eliminating the need to develop technical arguments for special cases. This is accomplished by establishing a general CLT for Qn,X with normalization assuming only 2+[delta] finite moments. Previous results for forms in dependent variables allowed only normalization with n1/2 and required at least four finite moments. Our technique uses approximations of Qn,X by a form Qn,Z in i.i.d. errors Z1,...,Zn. We develop sharp bounds for these approximations which in some cases are faster by the factor n1/2 compared to the existing results.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:117:y:2007:i:1:p:71-95
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    References listed on IDEAS

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    1. Mikosch, T., 1991. "Functional limit theorems for random quadratic forms," Stochastic Processes and their Applications, Elsevier, vol. 37(1), pages 81-98, February.
    2. 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.
    3. Kokoszka, P. & Mikosch, T., 1997. "The integrated periodogram for long-memory processes with finite or infinite variance," Stochastic Processes and their Applications, Elsevier, vol. 66(1), pages 55-78, February.
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    Cited by:

    1. Hira Koul & Donatas Surgailis & Nao Mimoto, 2015. "Minimum distance lack-of-fit tests under long memory errors," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(2), pages 119-143, February.
    2. Pavel Yaskov, 2018. "LLN for Quadratic Forms of Long Memory Time Series and Its Applications in Random Matrix Theory," Journal of Theoretical Probability, Springer, vol. 31(4), pages 2032-2055, December.
    3. Mynbayev, Kairat & Darkenbayeva, Gulsim, 2017. "Weak convergence of linear and quadratic forms and related statements on Lp-approximability," MPRA Paper 101686, University Library of Munich, Germany, revised Dec 2018.
    4. 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.
    5. Dmitrij Celov & Remigijus Leipus & Anne Philippe, 2010. "Asymptotic normality of the mixture density estimator in a disaggregation scheme," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(4), pages 425-442.
    6. Anne Philippe & Donata Puplinskaite & Donatas Surgailis, 2014. "Contemporaneous Aggregation Of Triangular Array Of Random-Coefficient Ar(1) Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(1), pages 16-39, January.
    7. Atchadé, Yves F. & Cattaneo, Matias D., 2014. "A martingale decomposition for quadratic forms of Markov chains (with applications)," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 646-677.
    8. Roueff, F. & Taqqu, M.S., 2009. "Central limit theorems for arrays of decimated linear processes," Stochastic Processes and their Applications, Elsevier, vol. 119(9), pages 3006-3041, September.
    9. Bhansali, R.J. & Giraitis, L. & Kokoszka, P.S., 2007. "Convergence of quadratic forms with nonvanishing diagonal," Statistics & Probability Letters, Elsevier, vol. 77(7), pages 726-734, April.

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