Convergence of quasi-stationary distributions in birth-death processes
type: Item
instance of: scholarly article
title: Convergence of quasi-stationary distributions in birth-death processes
zbMATH Open document ID: 0558.60066
DOI: 10.1016/0304-4149(84)90302-8
author: Julian Keilson
author: Ravi Ramaswamy
published in: Stochastic Processes and their Applications
publication date: 1984-01-01T00:00:00Z
review text: Let \({\mathcal N}(t)\) be an ergodic birth-death process on the positive half- lattice \({\mathcal N}\) and consider the sequence of partitions \({\mathcal N}=G_ k+B_ k\) where \(G_ k=(0,1,...,k)\). Let q(T,k) be the asymptotic distribution of the survivors on \(G_ k\) as \(t\to \infty\), the associated quasi-stationary distribution. Consider the reflecting process \({\mathcal N}^ R_ k(t)\) with reflecting barrier at k and two absorbing processes \({\mathcal N}^ 0_ k(t)\) and \({\mathcal N}^ Q_ k(t)\) where for the former, whenever a sample path reaches the absorbing state \(k+1\), it is replaced at 0 and for the latter it is replaced according to q(T,k). If \(e_ k^{RT}\), \(e_ k^{0T}\), and \(e_ k^{QT}\) are the corresponding ergodic distributions, then the authors show that \(e_ k^{QT}=q(T,k)\) and that \(e_ k^{0T}<e_ k^{QT}<e_ k^{RT}\) for all k where \(''<''\) stands for a partial order ''stochastically less than'' defined in an appendix to the paper. Since both \(e_ k^{0T}\) and \(e_ k^{RT}\) converge to \(e^ t\), the ergodic distribution of the original birth-death process, \(q(T,k)\to e^ T\) as \(t\to \infty\).
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 3890478
zbMATH Keywords: ergodic birth-death process
zbMATH Keywords: asymptotic distribution
zbMATH Keywords: quasi-stationary distribution
MaRDI profile type: Publication
full work available at URL: 0304-4149(84)90302-8
OpenAlex ID: W1970251133
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description: scientific article; zbMATH DE number 3890478