Geodesic
A generalization of the Mejzler–De Haan theorem
Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 1, pp. 177-189 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $(k_n)$ be a sequence of positive integers such that $k_n\to~\infty$ as $n\to\infty$. Let $X^\ast_{n1},\dots,X^\ast_{nk_n}$, $n\inN$, be a double array of random variables such that for each $n$ the random variables $X^\ast_{n1}\dots X^\ast_{nk_n}$ are independent with a common distribution function $F_n$, and let us denote $M^\ast_n=\max\{X^\ast_{n1},\dots,X^\ast_{nk_n}\}$. We consider an example of double array random variables connected with a certain combinatorial waiting time problem (including both dependent and independent cases), where $k_n=n$ for all $n$ and the limiting distribution function for $M^\ast_n$ is $\Lambda(x)=\exp(-e^{-x})$, although none of the distribution functions $F_n$ belongs to the domain of attraction $D(\Lambda)$. We also generalize the Mejzler–de Haan theorem and give the necessary and sufficient conditions for the sequence $(F_n)$ under which there exist sequences $a_n>0$ and $b_n\in R$, $n\inN$, such that $F_n^{k_n}(a_nx+b_n)\to\exp(-e^{-x})$ as $n\to\infty$ for every real $x$.
Keywords: extreme value distributions, double array, regular variation, double exponential distribution.
Mots-clés : domain of attraction 
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P. Mladenović. A generalization of the Mejzler–De Haan theorem. Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 1, pp. 177-189. http://geodesic.mathdoc.fr/item/TVP_2005_50_1_a12/

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