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Limit laws for extremes of dependent stationary Gaussian arrays

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  • Hashorva, Enkelejd
  • Weng, Zhichao

Abstract

In this paper we show that the componentwise maxima of weakly dependent bivariate stationary Gaussian triangular arrays converge in distribution after appropriate normalization to Hüsler–Reiss distribution. Under a strong dependence assumption, we prove that the limit distribution of the maxima is a mixture of a bivariate Gaussian distribution and Hüsler–Reiss distribution. An important new finding of our paper is that the componentwise maxima and componentwise minima remain asymptotically independent even in the settings of Hüsler and Reiss (1989) allowing further for weak dependence. Further we derive an almost sure limit theorem under the Berman condition for the components of the triangular array.

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  • Hashorva, Enkelejd & Weng, Zhichao, 2013. "Limit laws for extremes of dependent stationary Gaussian arrays," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 320-330.
    • Handle: RePEc:eee:stapro:v:83:y:2013:i:1:p:320-330
    DOI: 10.1016/j.spl.2012.09.017
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    References listed on IDEAS

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    1. Fahrner, I. & Stadtmüller, U., 1998. "On almost sure max-limit theorems," Statistics & Probability Letters, Elsevier, vol. 37(3), pages 229-236, March.
    2. Csáki, Endre & Gonchigdanzan, Khurelbaatar, 2002. "Almost sure limit theorems for the maximum of stationary Gaussian sequences," Statistics & Probability Letters, Elsevier, vol. 58(2), pages 195-203, June.
    3. Manjunath, B.G. & Frick, Melanie & Reiss, Rolf-Dieter, 2012. "Some notes on extremal discriminant analysis," Journal of Multivariate Analysis, Elsevier, vol. 103(1), pages 107-115, January.
    4. Hashorva, Enkelejd, 2005. "Elliptical triangular arrays in the max-domain of attraction of Hüsler-Reiss distribution," Statistics & Probability Letters, Elsevier, vol. 72(2), pages 125-135, April.
    5. Tan, Zhongquan & Peng, Zuoxiang, 2009. "Almost sure convergence for non-stationary random sequences," Statistics & Probability Letters, Elsevier, vol. 79(7), pages 857-863, April.
    6. Frick, Melanie & Reiss, Rolf-Dieter, 2010. "Limiting distributions of maxima under triangular schemes," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2346-2357, November.
    7. 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.
    8. Hüsler, Jürg & Reiss, Rolf-Dieter, 1989. "Maxima of normal random vectors: Between independence and complete dependence," Statistics & Probability Letters, Elsevier, vol. 7(4), pages 283-286, February.
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    Cited by:

    1. 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.
    2. Hashorva, Enkelejd & Peng, Liang & Weng, Zhichao, 2015. "Maxima of a triangular array of multivariate Gaussian sequence," Statistics & Probability Letters, Elsevier, vol. 103(C), pages 62-72.
    3. Enkelejd Hashorva & Zuoxiang Peng & Zhichao Weng, 2016. "Higher-order expansions of distributions of maxima in a Hüsler-Reiss model," Methodology and Computing in Applied Probability, Springer, vol. 18(1), pages 181-196, March.
    4. Das, Bikramjit & Engelke, Sebastian & Hashorva, Enkelejd, 2015. "Extremal behavior of squared Bessel processes attracted by the Brown–Resnick process," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 780-796.
    5. Tan, Zhongquan, 2013. "An almost sure limit theorem for the maxima of smooth stationary Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 83(9), pages 2135-2141.
    6. Withers, Christopher S. & Nadarajah, Saralees, 2015. "The joint distribution of the maximum and minimum of an AR(1) process," Statistics & Probability Letters, Elsevier, vol. 99(C), pages 77-84.

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