Almost sure convergence of extreme order statistics

Abstract

Let M_n^(k) denote the kth largest maximum of a sample (X₁,X₂,…,X_n) from parent X with continuous distribution. Assume there exist normalizing constants a_n>0, b_n∈ℝ and a nondegenerate distribution G such that $a_{n}^{-1}(M_{n}^{(1)}-b_{n})\stackrel{w}{\to}G$. Then for fixed k∈ℕ, the almost sure convergence of $$\begin{eqnarray*}\frac{1}{D_{N}}\sum_{n=k}^{N}d_{n}\mathbb{I}\{M_{n}^{(1)}\,{\le}\,a_{n}x_{1}\,{+}\,b_{n},M_{n}^{(2)}\,{\le}\,a_{n}x_{2}\,{+}\,b_{n},\ldots,M_{n}^{(k)}\le a_{n}x_{k}\,{+}\,b_{n}\}\end{eqnarray*}$$ is derived if the positive weight sequence (d_n) with D_N=∑_n=1^Nd_n satisfies conditions provided by Hörmann. Some practical issues of this result are also discussed.