Almost sure convergence of extreme order statistics

Almost sure convergence of extreme order statistics

Zuoxiang Peng, Jiaona Li, and Saralees Nadarajah

Source: Electron. J. Statist. Volume 3 (2009), 546-556.

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 $\begin{eqnarray*}\quad \qquad \frac{1}{D_{N}}\sum_{n=k}^{N}d_{n}\mathbb{I}\{M_{n}^{(1)}\,{\le}\,a_{n}x_{1}\,{+}\,b_{n},M_{n}^{(2)}\,{\le}\,a_{n}x_{2}\,{+}\,b_{n},\ldots,M_{n}^{(k)}\le a_{n}x_{k}\,{+}\,b_{n}\}\end{eqnarray*}$ 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.

First Page: Show

Primary Subjects: 62F15

Secondary Subjects: 60G70, 60F15

Keywords: Almost sure convergence; order statistics

Full-text: Open access

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.ejs/1245243980
Digital Object Identifier: doi:10.1214/08-EJS303
Mathematical Reviews number (MathSciNet): MR2519532

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