Asymptotic behavior of Markov evolutions until absorption
type: Item
instance of: scholarly article
title: Asymptotic behavior of Markov evolutions until absorption
zbMATH Open document ID: 0607.60062
DOI: 10.1007/BF01058490
author: V. V. Korolyuk
author: A. Tadzhiev
published in: Ukrainian Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: Let \(X_{\epsilon}(t)\) be a homogeneous Markov process with absorption state 0, defined by the Markov kernel \[ Q_{\epsilon}(x,A,t)=P_{\epsilon}(x,A)(1-e^{-\alpha (x)t}),\quad P_{\epsilon}(x,A)=P_ 0(x,A)-\epsilon P_ 1(x,A). \] Let \(y_{\epsilon}(t,x)\) be a family of homogeneous continuous processes with independent increments and uniformly bounded cumulants \(\psi (x,z)=\ln M(e^{izy(t,x)})/t.\) This paper gives an algorithm for the construction of an asymptotic expansion for the characteristic function of the Markov random evolution \(y_{\epsilon}(t)\) formed by \(y_{\epsilon}(t,x)\) and switched by \(X_{\epsilon}(t):\) \[ U_{\epsilon}(t)=e^{-\Lambda t}+\sum_{k\geq 1}\epsilon^ k[e^{-\Lambda t}U_ k(t)+V_ k(t/\epsilon)]. \] Where \(U_ k(t)\) and \(V_ k(t)\) are constructed recursively, \[ \Lambda =\int_{E}\pi (dx)[\alpha (x)P_ 1(x,E)-\psi (x,z)], \] \(\pi\) is a measure related to \(\alpha(x)\) and a stationary distribution of an unperturbed, uniformly ergodic Markov chain defined by \(P_ 0(x,\cdot)\).
Mathematics Subject Classification ID: 60J99
Mathematics Subject Classification ID: 60E10
Mathematics Subject Classification ID: 60F99
zbMATH DE Number: 3982204
zbMATH Keywords: asymptotic expansion for the characteristic function
zbMATH Keywords: stationary distribution
zbMATH Keywords: uniformly ergodic
MaRDI profile type: Publication
cites work: Q3935980
cites work: Central limit theorem for homogeneous processes with independent increments and semi-Markov commutations
cites work: Q4197802
cites work: Q3333839
full work available at URL: https://doi.org/10.1007/bf01058490
OpenAlex ID: W2085240538
description: scientific article; zbMATH DE number 3982204