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This article is cited in 6 scientific papers (total in 6 papers)
Short Communications
A generalization of the Mejzler–De Haan theorem
P. Mladenović University of Belgrade, Faculty of Mathematics
Abstract:
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, domain of attraction, regular variation, double exponential distribution.
Received: 16.09.2001
Citation:
P. Mladenović, “A generalization of the Mejzler–De Haan theorem”, Teor. Veroyatnost. i Primenen., 50:1 (2005), 177–189; Theory Probab. Appl., 50:1 (2006), 141–153
Linking options:
https://www.mathnet.ru/eng/tvp167https://doi.org/10.4213/tvp167 https://www.mathnet.ru/eng/tvp/v50/i1/p177
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