On multivariate extremal processes
type: Item
instance of: scholarly article
title: On multivariate extremal processes
zbMATH Open document ID: 0786.60070
author: Alexander Gnedin
published in: Journal of Multivariate Analysis
publication date: 1993-12-06T00:00:00Z
review text: Let \({\mathbf X}_{n,N}=(X^{(1)}_{n,N},\dots,X^{(m)}_{n,N})\), \(1 \leq n \leq N\), \(N \geq 1\), be a triangular array of \(m\)-dimensional random vectors. For each \(N\) assume that \({\mathbf X}_{1,N},\dots,{\mathbf X}_{N,N}\) are exchangeable and arrive at time points \(1/N\), \(2/N,\dots,1\), respectively. Now, for a Borel set \(B\) in \((0,1]\), let \(\xi_ N^{(l,k)}(B)=\#\{n:n/N \in B\) and \({\mathbf X}_{n,N}\) has a \(k\)-record in its \(l\)-th component\}. For a fixed \(K>1\), the paper obtains necessary and sufficient conditions for the convergence of the \(mK\)-variate process \(\{\xi_ N^{(l,k)}:1\leq l \leq m;\;1 \leq k \leq K\}\) to a Poisson process with independent components. These conditions are related to the one which states that the probability that the records are simultaneously broken at different components prior to time \(n/N\) approaches to 0 as \(n \to \infty\).
Mathematics Subject Classification ID: 60G70
zbMATH DE Number: 445126
zbMATH Keywords: record times
zbMATH Keywords: ranks
zbMATH Keywords: triangular array
zbMATH Keywords: exchangeable
zbMATH Keywords: Poisson process
zbMATH Keywords: records
reviewed by: H. N. Nagaraja
MaRDI profile type: Publication
full work available at URL: jmva.1993.1057
OpenAlex ID: W1973168314
DOI: 10.1006/JMVA.1993.1057
description: scientific article; zbMATH DE number 445126