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Application of copulas in the proof of the almost sure central limit theorem for the $k$th largest maxima of some random variables

Volume 45 / 2018

Marcin Dudziński, Konrad Furmańczyk Applicationes Mathematicae 45 (2018), 31-51 MSC: 60F15, 60F05, 60E05. DOI: 10.4064/am2340-10-2017 Published online: 2 March 2018

Abstract

Our aim is to prove the almost sure central limit theorem for the $k$th largest maxima $( M_{n}^{( k) }) $, $k=1,2,\ldots , $ of $X_{1},\ldots ,X_{n}$, $n \gt k$, where $( X_{i}) $ forms a stochastic process of identically distributed r.v.’s of continuous type, having a bounded, continuous density and such that, for any fixed $n$, the family $( X_{1},\ldots ,X_{n}) $ of r.v.’s has an Archimedean copula $C^{\varPsi }$ with the inverse function of its generator, $\varPsi ^{-1}$, satisfying the condition of complete monotonicity.

Authors

  • Marcin DudzińskiFaculty of Applied Informatics and Mathematics
    Department of Applied Mathematics
    Warsaw University of Life Sciences – SGGW
    Nowoursynowska 159
    02-776 Warszawa, Poland
    e-mail
  • Konrad FurmańczykFaculty of Applied Informatics and Mathematics
    Department of Applied Mathematics
    Warsaw University of Life Sciences – SGGW
    Nowoursynowska 159
    02-776 Warszawa, Poland
    e-mail

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