Boundary-value problems for sums of random variables, defined on a countably-valued Markov chain. II
type: Item
instance of: scholarly article
title: Boundary-value problems for sums of random variables, defined on a countably-valued Markov chain. II
zbMATH Open document ID: 0699.60069
DOI: 10.1007/BF01060534
author: N. S. Bratijchuk
published in: Ukrainian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: [For part I see the preceding entry, Zbl 0699.60068.] Let \((\xi_ n,x_ n)\) be a Markov additive process where \(x_ n\) is an infinite Markov chain and \(\xi_ n-\xi_{n-1}\) takes values in the set of integers. Assume that \[ P\{\xi_ 1-\xi_ 0\geq -1| x_ 0=i\}=1\quad and\quad P\{\xi_ 1-\xi_ 0=-1| x_ 0=i_ 0\}>0\quad for\quad some\quad i_ 0. \] Set \(\tau_ k=\inf \{n: \xi_ n>k\}\), \(\eta_ k=\xi_{\tau_ k}-k\). The main results of the article are connected with the asymptotic behaviour of the functions \[ (E\{e^{- \lambda \tau_ k/k^ 2},x_{\tau_ k}=j| x_ 0=i\}),\quad and\quad (P\{\eta_ K>\ell,x_{\tau_ k}=j| x_ 0=i\}) \] when \(k\to \infty\).
reviewed by: N. S. Bratijchuk
Mathematics Subject Classification ID: 60J50
Mathematics Subject Classification ID: 60J10
zbMATH DE Number: 4147249
zbMATH Keywords: boundary problem
zbMATH Keywords: Markov additive process
zbMATH Keywords: asymptotic behaviour
MaRDI profile type: Publication
cites work: Q5526485
full work available at URL: https://doi.org/10.1007/bf01060534
OpenAlex ID: W4205863108
description: scientific article; zbMATH DE number 4147249