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Limit distribution of the sum and maximum from multivariate Gaussian sequences

Author

Listed:
  • James, Barry
  • James, Kang
  • Qi, Yongcheng

Abstract

In this paper we study the asymptotic joint behavior of the maximum and the partial sum of a multivariate Gaussian sequence. The multivariate maximum is defined to be the coordinatewise maximum. Results extend univariate results of McCormick and Qi. We show that, under regularity conditions, if the maximum has a limiting distribution it is asymptotically independent of the partial sum. We also prove that the maximum of a stationary sequence, when normalized in a special sense which includes subtracting the sample mean, is asymptotically independent of the partial sum (again, under regularity conditions). The limiting distributions are also obtained.

Suggested Citation

  • James, Barry & James, Kang & Qi, Yongcheng, 2007. "Limit distribution of the sum and maximum from multivariate Gaussian sequences," Journal of Multivariate Analysis, Elsevier, vol. 98(3), pages 517-532, March.
    • Handle: RePEc:eee:jmvana:v:98:y:2007:i:3:p:517-532
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    References listed on IDEAS

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    1. Amram, Fred, 1985. "Multivariate extreme value distributions for stationary Gaussian sequences," Journal of Multivariate Analysis, Elsevier, vol. 16(2), pages 237-240, April.
    2. Wisniewski, Mateusz, 1996. "On extreme-order statistics and point processes of exceedances in multivariate stationary Gaussian sequences," Statistics & Probability Letters, Elsevier, vol. 29(1), pages 55-59, August.
    3. Hüsler, Jürg & Schüpbach, Michel, 1988. "Limit results for maxima in non-stationary multivariate Gaussian sequences," Stochastic Processes and their Applications, Elsevier, vol. 28(1), pages 91-99, April.
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    Cited by:

    1. Zhongquan Tan & Enkelejd Hashorva, 2014. "On Piterbarg Max-Discretisation Theorem for Standardised Maximum of Stationary Gaussian Processes," Methodology and Computing in Applied Probability, Springer, vol. 16(1), pages 169-185, March.
    2. Aiping Hu & Zuoxiang Peng & Yongcheng Qi, 2009. "Joint behavior of point process of exceedances and partial sum from a Gaussian sequence," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 70(3), pages 279-295, November.
    3. Jinhui Guo & Yingyin Lu, 2023. "Joint behavior of point processes of clusters and partial sums for stationary bivariate Gaussian triangular arrays," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(1), pages 17-37, February.
    4. Peng, Zuoxiang & Cao, Lunfeng & Nadarajah, Saralees, 2010. "Asymptotic distributions of maxima of complete and incomplete samples from multivariate stationary Gaussian sequences," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2641-2647, November.
    5. Gong, Siliang & Zhang, Kai & Liu, Yufeng, 2018. "Efficient test-based variable selection for high-dimensional linear models," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 17-31.

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