Ergodicity of an infinite dimensional renewal process
type: Item
instance of: scholarly article
title: Ergodicity of an infinite dimensional renewal process
zbMATH Open document ID: 0757.60085
DOI: 10.1016/0304-4149(92)90036-P
author: Enrique Daniel Andjel
author: Maria Eulalia Vares
published in: Stochastic Processes and their Applications
publication date: 1992-10-04T00:00:00Z
review text: A very special infinite-dimensional renewal process is constructed and studied. Processes of this type have been introduced by \textit{F. Spitzer} [Adv. Math., Suppl. Stud. 9, 147-155 (1986; Zbl 0608.60083)]. The state space of the process is \(\mathbb{R}^{\mathbb{Z}}_ +\). A state \(\eta=(\eta(k))_{k\in\mathbb{Z}}\) represents a vector of ages of objects. The \(k\)-th coordinate \(\eta(k)\) increases deterministically at rate 1 or jumps to 0 with failure rate \(c+{1\over 2}(\eta(k-1)+\eta(k+1))\) for some constant \(c\geq 0\). The main result states that this process is ergodic if and only if \(c>0\). The construction of the process requires a major part of the paper. It is modelled as a limit of Markov processes with finite- dimensional state spaces.
Mathematics Subject Classification ID: 60K05
zbMATH DE Number: 68094
zbMATH Keywords: multidimensional renewal processes
zbMATH Keywords: ergodicity
zbMATH Keywords: attractiveness
reviewed by: Helmut Wegmann
MaRDI profile type: Publication
cites work: Q3721531
cites work: Q5185817
cites work: Q4098445
cites work: Q4049667
cites work: Q3747450
description: scientific article; zbMATH DE number 68094