An extension of almost sure central limit theorem for order statistics

Abstract

Let {Xn,n ≥ 1} be a sequence of i.i.d. random variables. Let Mn and mn denote the first and the second largest maxima. Assume that there are normalizing sequences an > 0, bn and a nondegenerate limit distribution G, such that \(a_n^{-1}(M_n-b_n)\stackrel{d}{\rightarrow}G\). Assume also that {dk,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).