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Limit laws for extremes of dependent stationary Gaussian arrays

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  • Hashorva, Enkelejd
  • Weng, Zhichao
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    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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    Bibliographic Info

    Article provided by Elsevier in its journal Statistics & Probability Letters.

    Volume (Year): 83 (2013)
    Issue (Month): 1 ()
    Pages: 320-330

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    Handle: RePEc:eee:stapro:v:83:y:2013:i:1:p:320-330

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    Related research

    Keywords: Hüsler–Reiss distribution; Brown–Resnick copula; Gumbel max-domain of attraction; Berman condition; Almost sure limit theorem;

    References

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    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. Fahrner, I. & Stadtmüller, U., 1998. "On almost sure max-limit theorems," Statistics & Probability Letters, Elsevier, vol. 37(3), pages 229-236, March.
    6. Tan, Zhongquan & Peng, Zuoxiang, 2009. "Almost sure convergence for non-stationary random sequences," Statistics & Probability Letters, Elsevier, vol. 79(7), pages 857-863, April.
    7. 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.
    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. 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.

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