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Asymptotic normality of quadratic forms of martingale differences

Author

Listed:
  • Liudas Giraitis

    (Queen Mary University of London)

  • Masanobu Taniguchi

    (Waseda University)

  • Murad S. Taqqu

    (Boston University)

Abstract

We establish the asymptotic normality of a quadratic form $$Q_n$$ Q n in martingale difference random variables $$\eta _t$$ η t when the weight matrix A of the quadratic form has an asymptotically vanishing diagonal. Such a result has numerous potential applications in time series analysis. While for i.i.d. random variables $$\eta _t$$ η t , asymptotic normality holds under condition $$||A||_{sp}=o(||A||) $$ | | A | | s p = o ( | | A | | ) , where $$||A||_{sp}$$ | | A | | s p and ||A|| are the spectral and Euclidean norms of the matrix A, respectively, finding corresponding sufficient conditions in the case of martingale differences $$\eta _t$$ η t has been an important open problem. We provide such sufficient conditions in this paper.

Suggested Citation

  • Liudas Giraitis & Masanobu Taniguchi & Murad S. Taqqu, 2017. "Asymptotic normality of quadratic forms of martingale differences," Statistical Inference for Stochastic Processes, Springer, vol. 20(3), pages 315-327, October.
    • Handle: RePEc:spr:sistpr:v:20:y:2017:i:3:d:10.1007_s11203-016-9143-3
    DOI: 10.1007/s11203-016-9143-3
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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. Violetta Dalla & Liudas Giraitis & Hira L. Koul, 2014. "Studentizing Weighted Sums Of Linear Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(2), pages 151-172, March.
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    Cited by:

    1. A. V. Ivanov & N. N. Leonenko & I. V. Orlovskyi, 2020. "On the Whittle estimator for linear random noise spectral density parameter in continuous-time nonlinear regression models," Statistical Inference for Stochastic Processes, Springer, vol. 23(1), pages 129-169, April.
    2. Zhang, Tonglin, 2019. "General Gaussian estimation," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 234-247.

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