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Extremes of moving averages of random variables from the domain of attraction of the double exponential distribution

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  • Davis, Richard
  • Resnick, Sidney

Abstract

Let {Zn} be an iid sequence of random variables with common distribution F which belongs to the domain of attraction of exp{-e-x}. If in addition, F[epsilon]Sr([gamma]) (i.e.,limx-->[infinity] P[Z1+Z2>]/P[Z1>x]=d[epsilon](0, [infinity]) and , then it is shown that a point process based on the moving average process {Xn:=[Sigma][infinity]j=-[infinity]cjZn-j} converges weakly. A host of complementary results concerning extremal properties of {Xn} can then be derived from this convergence result. These include the convergence of maxima to extremal processes, the limit point process of exceedances, the joint limit distribution of the largest and second largest and the joint limit distribution of the largest and smallest. Convergence of a sequence of point processes based on the max-moving average process {V[infinity]n=-[infinity]cjZn-j} is also considered.

Suggested Citation

  • Davis, Richard & Resnick, Sidney, 1988. "Extremes of moving averages of random variables from the domain of attraction of the double exponential distribution," Stochastic Processes and their Applications, Elsevier, vol. 30(1), pages 41-68, November.
    • Handle: RePEc:eee:spapps:v:30:y:1988:i:1:p:41-68
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    Cited by:

    1. Geluk, J.L. & De Vries, C.G., 2006. "Weighted sums of subexponential random variables and asymptotic dependence between returns on reinsurance equities," Insurance: Mathematics and Economics, Elsevier, vol. 38(1), pages 39-56, February.
    2. Hashorva, Enkelejd, 2015. "Extremes of aggregated Dirichlet risks," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 334-345.
    3. Li, Jinzhu, 2022. "Asymptotic results on marginal expected shortfalls for dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 102(C), pages 146-168.
    4. Yang, Yang & Hashorva, Enkelejd, 2013. "Extremes and products of multivariate AC-product risks," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 312-319.
    5. Tyran-Kaminska, Marta, 2010. "Convergence to Lévy stable processes under some weak dependence conditions," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1629-1650, August.
    6. Tang, Qihe & Yang, Fan, 2012. "On the Haezendonck–Goovaerts risk measure for extreme risks," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 217-227.
    7. Balakrishnan, N. & Hashorva, E., 2013. "Scale mixtures of Kotz–Dirichlet distributions," Journal of Multivariate Analysis, Elsevier, vol. 113(C), pages 48-58.
    8. Fasen, Vicky, 2006. "Extremes of subexponential Lévy driven moving average processes," Stochastic Processes and their Applications, Elsevier, vol. 116(7), pages 1066-1087, July.
    9. Kühne Robert & Rüschendorf Ludger, 2003. "Optimal stopping and cluster point processes," Statistics & Risk Modeling, De Gruyter, vol. 21(3), pages 261-282, March.

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