Boundary crossing random variables related to quantile convergence
type: Item
instance of: scholarly article
title: Boundary crossing random variables related to quantile convergence
zbMATH Open document ID: 0634.60027
DOI: 10.1016/0167-7152(87)90084-8
author: Ralph P. Russo
author: Chern-Ching Chao
published in: Statistics \& Probability Letters
publication date: 1987-01-01T00:00:00Z
review text: Let \(X_ 1,X_ 2,..\). be a sequence of independent random variables with distribution F. Suppose that \(0<p<1\), that \(\xi_ p\) is the unique p-th quantile of F, and that \(\xi_{p,n}\) is the sample p-th quantile of \(X_ 1,...,X_ n\). If \(b(n)\to 0^+\) sufficiently slowly then \[ N(b(n))=\sum^{\infty}_{n=1}I\{| \xi_{p,n}-\xi_ p| >b(n)\}\quad and \] \[ L(b(n))=\sup \{n:| \xi_{p,n}-\xi_ p| >b(n)\} \] are ty space and (\({\mathcal F}_ n)\) an increasing family of sub-\(\sigma\)-algebras of \({\mathcal F}\) with supremum \({\mathcal F}_{\infty}\). If \(\mu\) : \({\mathcal F}\to {\mathbb{R}}\) is a signed measure, \(\mu\) induces an (\({\mathcal F}_ n)_{n\in {\mathbb{N}}\cup \{\infty \}}\)-adapted process \((X_ n)\) in the following way. Let \(\mu_ n\) denote the restriction of \(\mu\) to \({\mathcal F}_ n\) and define \(X_ n\) to be the Radon-Nikodym derivative of the absolutely continuous part of \(\mu_ n\) relative to \(\lambda\). The Andersen-Jessen theorem states that lim \(X_ n=X_{\infty}^ a.\)s.. This result has been extended to vector-valued stochastic processes induced by finitely additive set functions or set function sequences. In the present paper the author studies in a quite general context structure and limit behaviour of such induced processes. He shows how the Andersen-Jessen theorem and its extensions can be located within the convergence theory for martingales and their generalizations.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G17
zbMATH DE Number: 4032705
zbMATH Keywords: sample quantiles
zbMATH Keywords: boundary crossings
zbMATH Keywords: law of the iterated logarithm for quantiles
zbMATH Keywords: Andersen-Jessen theorem
zbMATH Keywords: vector-valued stochastic processes induced by finitely additive set functions
zbMATH Keywords: convergence theory for martingales
reviewed by: Egbert Dettweiler
MaRDI profile type: Publication
full work available at URL: 0167-7152(87)90084-8
OpenAlex ID: W2051713153
cites work: On sample quantiles from a regularly varying distribution function
cites work: Some limit results for lag sums of independent, non-i.i.d., random variables
cites work: On the last time and the number of boundary crossings related to the strong law of large numbers and the law of the iterated logarithm
cites work: Q3857500
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description: scientific article; zbMATH DE number 4032705