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A Limit Theorem For Quadratic Forms And Its Applications

Author

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  • Wu, Wei Biao
  • Shao, Xiaofeng

Abstract

We consider quadratic forms of martingale differences and establish a central limit theorem under mild and easily verifiable conditions. By approximating Fourier transforms of stationary processes by martingales, our central limit theorem is applied to the smoothed periodogram estimate of spectral density functions. Our results go beyond earlier ones by allowing a variety of nonlinear time series and by avoiding strong mixing and/or summability conditions on joint cumulants.We thank the two reviewers for their detailed comments, which led to substantial improvements. The work is supported in part by NSF grant DMS-0478704.

Suggested Citation

  • Wu, Wei Biao & Shao, Xiaofeng, 2007. "A Limit Theorem For Quadratic Forms And Its Applications," Econometric Theory, Cambridge University Press, vol. 23(5), pages 930-951, October.
    • Handle: RePEc:cup:etheor:v:23:y:2007:i:05:p:930-951_07
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    Cited by:

    1. Zhongjun Qu, 2011. "A Test Against Spurious Long Memory," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 29(3), pages 423-438, July.
    2. 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.
    3. Jiti Gao & Bin Peng & Yayi Yan, 2021. "Parameter Stability Testing for Multivariate Dynamic Time-Varying Models," Monash Econometrics and Business Statistics Working Papers 11/21, Monash University, Department of Econometrics and Business Statistics.
    4. 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.
    5. Jirak, Moritz, 2012. "Change-point analysis in increasing dimension," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 136-159.
    6. 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.
    7. He, Yi & Jaidee, Sombut & Gao, Jiti, 2023. "Most powerful test against a sequence of high dimensional local alternatives," Journal of Econometrics, Elsevier, vol. 234(1), pages 151-177.
    8. Zhengyan Lin & Hanchao Wang, 2016. "On Convergence to Stochastic Integrals," Journal of Theoretical Probability, Springer, vol. 29(3), pages 717-736, September.
    9. Jirak, Moritz, 2013. "A Darling–Erdös type result for stationary ellipsoids," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 1922-1946.
    10. 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.
    11. Zhang, Tonglin, 2019. "General Gaussian estimation," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 234-247.
    12. Zhang, Rong-Mao & Sin, Chor-yiu (CY) & Ling, Shiqing, 2015. "On functional limits of short- and long-memory linear processes with GARCH(1,1) noises," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 482-512.
    13. Yi He & Sombut Jaidee & Jiti Gao, 2020. "Most Powerful Test against High Dimensional Free Alternatives," Monash Econometrics and Business Statistics Working Papers 13/20, Monash University, Department of Econometrics and Business Statistics.
    14. Ángel Felipe & María Jaenada & Pedro Miranda & Leandro Pardo, 2023. "Restricted Distance-Type Gaussian Estimators Based on Density Power Divergence and Their Applications in Hypothesis Testing," Mathematics, MDPI, vol. 11(6), pages 1-41, March.
    15. Kin Wai Chan & Chun Yip Yau, 2017. "High-order Corrected Estimator of Asymptotic Variance with Optimal Bandwidth," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(4), pages 866-898, December.

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