Almost Sure Convergence of Extreme Order Statistics

TL;DR

This paper establishes almost sure convergence results for extreme order statistics of samples from continuous distributions, extending classical convergence in distribution to almost sure convergence under specific weighting conditions.

Contribution

It provides new almost sure convergence theorems for the joint behavior of top order statistics, generalizing existing results with weighted sums and conditions from Hörmann.

Findings

Almost sure convergence of weighted sums of order statistics

Conditions on weights for convergence are specified

Extends classical distributional convergence to almost sure convergence

Abstract

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.