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Extremes of subexponential Lévy driven moving average processes

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  • Fasen, Vicky

Abstract

In this paper we study the extremal behavior of a stationary continuous-time moving average process for , where f is a deterministic function and L is a Lévy process whose increments, represented by L(1), are subexponential and in the maximum domain of attraction of the Gumbel distribution. We give necessary and sufficient conditions for Y to be a stationary, infinitely divisible process, whose stationary distribution is subexponential, and in this case we calculate its tail behavior. We show that large jumps of the Lévy process in combination with extremes of f cause excesses of Y and thus properly chosen discrete-time points are sufficient for specifying the extremal behavior of the continuous-time process Y. We describe the extremal behavior of Y completely as a weak limit of marked point processes. A complementary result guarantees the convergence of running maxima of Y to the Gumbel distribution.

Suggested Citation

  • Fasen, Vicky, 2006. "Extremes of subexponential Lévy driven moving average processes," Stochastic Processes and their Applications, Elsevier, vol. 116(7), pages 1066-1087, July.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:7:p:1066-1087
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    References listed on IDEAS

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    1. 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.
    2. Cline, D. B. H. & Samorodnitsky, G., 1994. "Subexponentiality of the product of independent random variables," Stochastic Processes and their Applications, Elsevier, vol. 49(1), pages 75-98, January.
    3. Gushchin, Alexander A. & Küchler, Uwe, 2000. "On stationary solutions of delay differential equations driven by a Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 88(2), pages 195-211, August.
    4. Braverman, Michael & Samorodnitsky, Gennady, 1995. "Functionals of infinitely divisible stochastic processes with exponential tails," Stochastic Processes and their Applications, Elsevier, vol. 56(2), pages 207-231, April.
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