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An extension of almost sure central limit theorem for order statistics

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Abstract

Let {X n ,n ≥ 1} be a sequence of i.i.d. random variables. Let M n and m n denote the first and the second largest maxima. Assume that there are normalizing sequences a n  > 0, b n and a nondegenerate limit distribution G, such that \(a_n^{-1}(M_n-b_n)\stackrel{d}{\rightarrow}G\). Assume also that {d k ,k ≥ 1} are positive weights obeying some mild conditions. Then for x > y we have

$$ \lim\limits_{n\rightarrow\infty}\!\frac{1}{D_n}\!\sum\limits_{k=2}^{n}d_kI\left\{\!\frac{M_k\!-\!b_k}{a_k}\!\leq\! x, \frac{m_k\!-\!b_k}{a_k}\!\leq\! y\!\right\}\!=\!G(y)\left\{\log G(x)\!-\!\log G(y)\!+\!1\right\}\,\,a.s. $$

when G(y) > 0 (and to zero when G(y) = 0).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

    Tong Bin & Peng Zuoxiang

  2. School of Mathematics, University of Manchester, Manchester, UK

    Saralees Nadarajah

Authors
  1. Tong Bin
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  2. Peng Zuoxiang
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  3. Saralees Nadarajah
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Correspondence to Saralees Nadarajah.

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Bin, T., Zuoxiang, P. & Nadarajah, S. An extension of almost sure central limit theorem for order statistics. Extremes 12, 201–209 (2009). https://doi.org/10.1007/s10687-008-0075-1

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  • DOI: https://doi.org/10.1007/s10687-008-0075-1

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