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Large deviations for quadratic forms of stationary Gaussian processes

Author

Listed:
  • Bercu, B.
  • Gamboa, F.
  • Rouault, A.

Abstract

A large deviation principle is proved for Toeplitz quadratic forms of centred stationary Gaussian processes. The rate function is obtained by a sharp study of the behaviour of eigenvalues of a product of two Toeplitz matrices. Some statistical applications such as the likelihood ratio test and the estimation of the parameter of an autoregressive Gaussian process are also provided.

Suggested Citation

  • Bercu, B. & Gamboa, F. & Rouault, A., 1997. "Large deviations for quadratic forms of stationary Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 71(1), pages 75-90, October.
    • Handle: RePEc:eee:spapps:v:71:y:1997:i:1:p:75-90
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    References listed on IDEAS

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    1. Bryc, Wlodzimierz & Smolenski, Wlodzimierz, 1993. "On the large deviation principle for a quadratic functional of the autoregressive process," Statistics & Probability Letters, Elsevier, vol. 17(4), pages 281-285, July.
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    Cited by:

    1. Macci, Claudio & Pacchiarotti, Barbara, 2017. "Large deviations for estimators of the parameters of a neuronal response latency model," Statistics & Probability Letters, Elsevier, vol. 126(C), pages 65-75.
    2. Kakizawa, Yoshihide, 2007. "Moderate deviations for quadratic forms in Gaussian stationary processes," Journal of Multivariate Analysis, Elsevier, vol. 98(5), pages 992-1017, May.
    3. Yu, Miao & Si, Shen, 2009. "Moderate deviation principle for autoregressive processes," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 1952-1961, October.
    4. Worms, Julien, 2001. "Large and moderate deviations upper bounds for the Gaussian autoregressive process," Statistics & Probability Letters, Elsevier, vol. 51(3), pages 235-243, February.
    5. Yu Miao & Yanling Wang & Guangyu Yang, 2015. "Moderate Deviation Principles for Empirical Covariance in the Neighbourhood of the Unit Root," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(1), pages 234-255, March.
    6. Gamboa, F. & Rouault, A. & Zani, M., 1999. "A functional large deviations principle for quadratic forms of Gaussian stationary processes," Statistics & Probability Letters, Elsevier, vol. 43(3), pages 299-308, July.
    7. Ginovyan, Mamikon S. & Sahakyan, Artur A., 2013. "On the trace approximations of products of Toeplitz matrices," Statistics & Probability Letters, Elsevier, vol. 83(3), pages 753-760.
    8. Zani, Marguerite, 2002. "Large Deviations for Quadratic Forms of Locally Stationary Processes," Journal of Multivariate Analysis, Elsevier, vol. 81(2), pages 205-228, May.
    9. Kakizawa, Yoshihide, 2000. "On Bahadur asymptotic efficiency of the maximum likelihood and quasi-maximum likelihood estimators in Gaussian stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 85(1), pages 29-44, January.
    10. Kley, Tobias & Preuss, Philip & Fryzlewicz, Piotr, 2019. "Predictive, finite-sample model choice for time series under stationarity and non-stationarity," LSE Research Online Documents on Economics 101748, London School of Economics and Political Science, LSE Library.
    11. Kanaya, Shin & Otsu, Taisuke, 2012. "Large deviations of realized volatility," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 546-581.
    12. Hacène Djellout & Arnaud Guillin & Yacouba Samoura, 2014. "Large Deviations Of The Realized (Co-)Volatility Vector," Working Papers hal-01082903, HAL.
    13. Hui, Jiang, 2010. "Moderate deviations for estimators of quadratic variational process of diffusion with compound Poisson jumps," Statistics & Probability Letters, Elsevier, vol. 80(17-18), pages 1297-1305, September.
    14. Zani, Marguerite, 2002. "Large deviations for squared radial Ornstein-Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 102(1), pages 25-42, November.
    15. Djellout, Hacène & Guillin, Arnaud & Samoura, Yacouba, 2017. "Estimation of the realized (co-)volatility vector: Large deviations approach," Stochastic Processes and their Applications, Elsevier, vol. 127(9), pages 2926-2960.
    16. Miao, Yu & Yin, Qing, 2024. "Cramér’s moderate deviations for the LS estimator of the autoregressive processes in the neighborhood of the unit root," Statistics & Probability Letters, Elsevier, vol. 209(C).
    17. Hacène Djellout & Arnaud Guillin & Yacouba Samoura, 2017. "Large Deviations Of The Realized (Co-)Volatility Vector," Post-Print hal-01082903, HAL.

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