Almost Sure Convergence of Extreme Order Statistics

• Published in: arXiv.org
• Main Authors: Peng, Zuoxiang, Li, Jiaona, Nadarajah, Saralees
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 03.10.2008
• Subjects: Convergence; Extreme values; Mathematics - Statistics Theory; Normalizing; Statistics - Theory
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.0810.0579

Let \(M_n^{(k)}\) denote the \(k\)th largest maximum of a sample \((X_1,X_2,...,X_n)\) from parent \(X\) with continuous distribution. Assume there exist normalizing constants \(a_n>0\), \(b_n\in \mathbb{R}\) and a nondegenerate distribution \(G\) such that \(a_n^{-1}(M_n^{(1)}-b_n)\stackrel{w}{\to}G\). Then for fixed \(k\in \mathbb{N}\), the almost sure convergence of \[\frac{1}{D_N}\sum_{n=k}^Nd_n\mathbb{I}\{M_n^{(1)}\le a_nx_1+b_n,M_n^{(2)}\le a_nx_2+b_n,...,M_n^{(k)}\le a_nx_k+b_n\}\] is derived if the positive weight sequence \((d_n)\) with \(D_N=\sum_{n=1}^Nd_n\) satisfies conditions provided by H\"{o}rmann.