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An extension of almost sure central limit theorem for order statistics

Authors:

Tong Bin Peng Zuoxiang Saralees Nadarajah

Institution:

(1) School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China;(2) School of Mathematics, University of Manchester, Manchester, UK

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).

Keywords:

Almost sure central limit theorems Extreme order statistics Summation methods

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