Abstract
We study the limit behavior of power sums and norms of i.i.d. positive samples from the max domain of attraction of an extreme value distribution. To this end, we combine limit theorems for sums and for maxima and use a link between extreme value theory and the Lévy measures of certain infinitely divisible laws, which are limit distributions of power sums. In connection with the von Mises representation of the Gumbel max domain of attraction, this new approach allows us to extend the limit results for power sums found in Ben Arous et al. (Probab Theory Relat Fields 132:579–612, 2005) and Bogachev (J Theor Probab 19:849–873, 2006). Furthermore, our findings shed a new light on the results of Schlather (Ann Probab 29:862–881, 2001) and treat the Gumbel case which is missing there.
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Acknowledgments
The author is grateful to Martin Schlather and Zakhar Kabluchko for hints and helpful discussions and would like to thank an anonymous referee for very valuable comments.
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The author has been financially supported by the Georg Lichtenberg programm “Applied Statistics & Empirical Methods”.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Janßen, A. Limit laws for power sums and norms of i.i.d. samples. Probab. Theory Relat. Fields 146, 515–533 (2010). https://doi.org/10.1007/s00440-008-0198-y
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DOI: https://doi.org/10.1007/s00440-008-0198-y
Keywords
- Central limit theorem
- Extreme value theory
- Infinitely divisible distributions
- l p -norms
- Power sums
- Stable distributions
- von Mises representation