Limit laws for power sums and norms of i.i.d. samples
type: Item
instance of: scholarly article
title: Limit laws for power sums and norms of i.i.d. samples
zbMATH Open document ID: 1191.60026
publication date: 2010-02-09T00:00:00Z
review text: For i.i.d. positive valued random variables \(X_1, \dots, X_n,\) the author studies the limit behaviour of \(l_{p(n)}\)-norms of samples \[ \| X_{1n}\|_{p(n)} = \left( \sum_{i=1}^{n} X_i^{p(n)} \right)^{1 / p(n)},\quad p(n) > 0, \] where \(p(n)\) grows with \(n\), and the related behaviour of power sums of the form \((\| X_{1n}\|_{p(n)})^{p(n)} = \sum_{i=1}^{n}X_i^{p(n)}\). It is shown that nontrivial limit laws arise if \(p(n)\) is chosen in accordance with the tail behaviour of \(X_1\). The limit behaviour of power sums and norms of samples is shown to be governed by max domains of attraction of extreme value theory by establishing a link between extreme value theory and the Lévy measures of certain infinitely divisible laws, which are limit distributions of power sums. Using the von Mises representation for the Gumbel max domain of attraction, the results are extended to the limit results for power sums found in [\textit{G. Ben Arous, L. V. Bogachev} and \textit{S. A. Molchanov}, Probab. Theory Relat. Fields 132, No.~4, 579--612 (2005; Zbl 1073.60017)] and [\textit{L. Bogachev}, J. Theor. Probab. 19, No.~4, 849--873 (2006; Zbl 1128.60016)]. The approach also simplifies some of the proofs in [\textit{M. Schlather}, Ann. Probab. 29, No.~2, 862--881 (2001; Zbl 1014.60015)].
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 5668293
zbMATH Keywords: extreme value theory
zbMATH Keywords: central limit theorem
zbMATH Keywords: infinitely divisible laws
zbMATH Keywords: stable laws
zbMATH Keywords: Levy measures
zbMATH Keywords: \(l_p\) norms
zbMATH Keywords: power sums
zbMATH Keywords: von Mises representation
author: Anja Janssen
reviewed by: Sreenivasan Ravi
MaRDI profile type: Publication
full work available at URL: s00440-008-0198-y
OpenAlex ID: W2170151701
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DOI: 10.1007/S00440-008-0198-Y
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 5668293