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A connection between extreme value theory and long time approximation of SDEs

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  • Panloup, Fabien

Abstract

We consider a sequence ([xi]n)n>=1 of i.i.d. random values residing in the domain of attraction of an extreme value distribution. For such a sequence, there exist (an) and (bn), with an>0 and for every n>=1, such that the sequence (Xn) defined by Xn=(max([xi]1,...,[xi]n)-bn)/an converges in distribution to a non-degenerated distribution. In this paper, we show that (Xn) can be viewed as an Euler scheme with a decreasing step of an ergodic Markov process solution to a SDE with jumps and we derive a functional limit theorem for the sequence (Xn) from some methods used in the long time numerical approximation of ergodic SDEs.

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  • Panloup, Fabien, 2009. "A connection between extreme value theory and long time approximation of SDEs," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3583-3607, October.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:10:p:3583-3607
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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. Basak, Gopal K. & Hu, Inchi & Wei, Ching-Zong, 1997. "Weak convergence of recursions," Stochastic Processes and their Applications, Elsevier, vol. 68(1), pages 65-82, May.
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