On the rate of convergence of spatial birth-and-death processes
type: Item
instance of: scholarly article
title: On the rate of convergence of spatial birth-and-death processes
zbMATH Open document ID: 0702.60072
DOI: 10.1007/BF00050669
author: Jesper Møller
published in: Annals of the Institute of Statistical Mathematics
publication date: 1989-01-01T00:00:00Z
review text: A spatial birth-and-death process is a continuous-time Markov chain X(t) on a state space \(\Omega =\cup^{\infty}_{0}\Omega_ n\), where the \(\Omega_ n\) are disjoint and \(\Omega_ 0\) is a singleton. When the process is at \(x\in \Omega_ n\) then at the next transition it can only move to a state in \(\Omega_{n+1}\) or \(\Omega_{n-1}\) (at rates \(\beta\) (x), \(\delta\) (x) respectively). Let \(Q_ t(x,F)=P(X(t)\in F | X(0)=x)\) denote the transition distribution, for appropriate measurable subsets F of \(\Omega\). The author studies sufficient conditions for the distribution to converge geometrically fast to the invariant distribution. These results follow from conditions which ensure that \[ \sup_{F}| \int Q_ t(x,F)\gamma (dx)-\int Q_ t(y,F)\kappa (dy)| \] converges to zero geometrically fast, for suitable initial distributions \(\gamma\) and \(\kappa\).
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60J25
zbMATH DE Number: 4151560
zbMATH Keywords: invariant distribution
zbMATH Keywords: geometrical convergence
zbMATH Keywords: spatial birth-and-death process
reviewed by: D. P. Kennedy
describes a project that uses: spatial
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00050669
OpenAlex ID: W1504223551
description: scientific article; zbMATH DE number 4151560