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 Teor. Veroyatnost. i Primenen., 2005, Volume 50, Issue 1, Pages 177–189 (Mi tvp167)

Short Communications

A generalization of the Mejzler–De Haan theorem

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 $X^\ast_{n1},…,X^\ast_{nk_n}$, $n\inN$, be a double array of random variables such that for each $n$ the random variables $X^\ast_{n1}…X^\ast_{nk_n}$ are independent with a common distribution function $F_n$, and let us denote $M^\ast_n=\max\{X^\ast_{n1},…,X^\ast_{nk_n}\}$. 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.

DOI: https://doi.org/10.4213/tvp167

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English version:
Theory of Probability and its Applications, 2006, 50:1, 141–153

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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

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp167
• https://doi.org/10.4213/tvp167
• http://mi.mathnet.ru/eng/tvp/v50/i1/p177

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This publication is cited in the following articles:
1. Mladenović P., “On extreme steps of a random function on finite sets”, Indian J. Pure Appl. Math., 37:2 (2006), 89–98
2. Mladenović P., “Limit distributions for the problem of collecting pairs”, Bernoulli, 14:2 (2008), 419–439
3. Mladenović P., Vukmirović J., “Rates of convergence in certain limit theorem for extreme values”, J. Math. Anal. Appl., 363:1 (2010), 287–295
4. Jockovic J., Mladenovic P., “Coupon collector's problem and generalized Pareto distributions”, J Statist Plann Inference, 141:7 (2011), 2348–2352
5. Glavas L., Mladenovic P., “New Limit Results Related to the Coupon Collector'S Problem”, Stud. Sci. Math. Hung., 55:1 (2018), 115–140
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