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Gumbel and Fréchet convergence of the maxima of independent random walks

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  29 April 2020

Thomas Mikosch  and
Jorge Yslas Open the ORCID record for Jorge Yslas [Opens in a new window]
Thomas Mikosch*
Affiliation:
University of Copenhagen
Jorge Yslas*
Affiliation:
University of Copenhagen
*
*Postal address: Department of Mathematics, Universitetsparken 5, DK-2100Copenhagen, Denmark.
*Postal address: Department of Mathematics, Universitetsparken 5, DK-2100Copenhagen, Denmark.
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Abstract

We consider point process convergence for sequences of independent and identically distributed random walks. The objective is to derive asymptotic theory for the largest extremes of these random walks. We show convergence of the maximum random walk to the Gumbel or the Fréchet distributions. The proofs depend heavily on precise large deviation results for sums of independent random variables with a finite moment generating function or with a subexponential distribution.

Keywords

Large deviationsubexponential distributionregular variationextreme value theoryGumbel distributionFréchet distributionmaximum random walk

MSC classification

Primary: 60F10: Large deviations
Secondary: 60F05: Central limit and other weak theorems 60G50: Sums of independent random variables; random walks 60G55: Point processes 60G70: Extreme value theory; extremal processes

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 1 , March 2020 , pp. 213 - 236
Copyright
© Applied Probability Trust 2020

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