Limit laws for power sums and norms of i.i.d. samples
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)].
scientific article; zbMATH DE number 5668293
scientific article