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Convergence to type I distribution of the extremes of sequences defined by random difference equation

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  • Hitczenko, Pawel

Abstract

We study the extremes of a sequence of random variables (Rn) defined by the recurrence Rn=MnRn-1+q, n>=1, where R0 is arbitrary, (Mn) are iid copies of a non-degenerate random variable M, 0 0 is a constant. We show that under mild and natural conditions on M the suitably normalized extremes of (Rn) converge in distribution to a double-exponential random variable. This partially complements a result of de Haan, Resnick, Rootzén, and de Vries who considered extremes of the sequence (Rn) under the assumption that .

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  • Hitczenko, Pawel, 2011. "Convergence to type I distribution of the extremes of sequences defined by random difference equation," Stochastic Processes and their Applications, Elsevier, vol. 121(10), pages 2231-2242, October.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:10:p:2231-2242
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    References listed on IDEAS

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    1. Devroye, Luc & Fawzi, Omar, 2010. "Simulating the Dickman distribution," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 242-247, February.
    2. de Haan, Laurens & Resnick, Sidney I. & Rootzén, Holger & de Vries, Casper G., 1989. "Extremal behaviour of solutions to a stochastic difference equation with applications to arch processes," Stochastic Processes and their Applications, Elsevier, vol. 32(2), pages 213-224, August.
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