Almost sure convergence of sample range
type: Item
instance of: scholarly article
title: Almost sure convergence of sample range
zbMATH Open document ID: 1164.60018
author: Tan Zhongquan
author: Peng Zuoxiang
author: Saralees Nadarajah
published in: Extremes
publication date: 2009-02-28T00:00:00Z
review text: An almost sure limit theorem is derived for the sample range statistics \( R_n=\max_{1\leq i\leq n} X_i-\min_{1\leq i\leq n} X_i \), where \(X_i\) are i.i.d. r.v.s. It states that if for some nonrandom \(\alpha_n\), \(\beta_n\) and some nondegenerate CDF \(G\) \(\zeta_n=\alpha_n(R_n-\beta_n)\) converges in distribution to \(G\) then at any point \(x\) of continuity of \(G\) \[ \lim_{n\to\infty}{1\over\log n}\sum_{k=1}^n {1\over k}{\mathbf 1}_{\{\zeta_k\leq x\}}=G, \text{a.s.} \] and \[ \lim_{n\to\infty}{1\over\log n}\sum_{k=1}^n {1\over k} f(\zeta_k)=\int_{-\infty}^{+\infty} f(x)dG(x) \] for any bounded Lipschitz function \(f\).
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 62G30
zbMATH DE Number: 5520591
zbMATH Keywords: almost sure central limit theorem
zbMATH Keywords: max domain of attraction
zbMATH Keywords: asymptotic distribution
MaRDI profile type: Publication
full work available at URL: s10687-007-0042-2
OpenAlex ID: W2344642895
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DOI: 10.1007/S10687-007-0042-2
description: scientific article; zbMATH DE number 5520591