Records in a partially ordered set
type: Item
instance of: scholarly article
title: Records in a partially ordered set
zbMATH Open document ID: 0678.60001
author: Charles M. Goldie
author: Sidney I. Resnick
published in: The Annals of Probability
publication date: 1989-01-01T00:00:00Z
review text: Let S be a set equipped with a strict partial order \(``<''\) and let \(X_ 1,X_ 2,..\). be a sequence of independent, identically distributed S- valued random elements; \(X_ n\) is said to be a record iff \(X_ k<X_ n\), \(k=1,2,...,n-1.\) The authors define a hazard measure H on (S,\(s)\), where \(s\) is a \(\sigma\)-algebra of subsets of S. H(A), \(A\in s\), is the expected number of records that fall into A. The number \(N_ A\) of records in A is finite a.s. or infinite a.s. according as H(A) is finite or infinite. In the former case the authors calculate all the moments of \(N_ A\) and the probability law of \(N_ A\). They also show that the sequence of record values in A is a Markov chain and that for several sets A,B,... that are totally ordered the restrictions of the records process to the individual sets form independent processes. In the particular case of \(R^ 2\) the authors apply the general theory to the case of probability laws in the domain of attraction of a bivariate extreme-value law.
Mathematics Subject Classification ID: 60B05
Mathematics Subject Classification ID: 60K99
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 4111664
zbMATH Keywords: hazard measure
zbMATH Keywords: records process
zbMATH Keywords: domain of attraction
zbMATH Keywords: bivariate extreme- value law
reviewed by: Wiesław Dziubdziela
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991421
OpenAlex ID: W2072651582
DOI: 10.1214/AOP/1176991421
description: scientific article; zbMATH DE number 4111664