On the distribution of time spent by a Markov chain at different levels until achieving a fixed state
type: Item
instance of: scholarly article
zbMATH Open document ID: 1247.60108
DOI: 10.1137/S0040585X97985285
author: Ya. A. Lyulko
publication date: 2012-05-07T00:00:00Z
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 6031480
zbMATH Keywords: Markov chain
zbMATH Keywords: residence time
zbMATH Keywords: geometric distribution
zbMATH Keywords: skew random walk
zbMATH Keywords: skew Brownian motion
zbMATH Keywords: local time
zbMATH Keywords: Donsker-Prokhorov invariance principle
MaRDI profile type: Publication
full work available at URL: s0040585x97985285
OpenAlex ID: W1964455158
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title: On the distribution of time spent by a Markov chain at different levels until achieving a fixed state
published in: Theory of Probability and its Applications
review text: The author considers a homogeneous Markov chain with discrete time and countable state space. The main problem is to find the distribution of the number of crossings of the level \(a\) up to the first hitting time to the state \(b\). It is named the distribution of the residence time. It is proved that the distribution of the residence time is geometric with some unknown parameters. The main result is based on the fact that the parts of Markov chain between two hittings of the same state are independent. The unknown parameter are calculated for the skew random walk. In the second part of the paper the weak limit theorem is obtained for for passage from a residence time to a local time of the skew Brownian motion. It is established that the corresponding local time of the skew Brownian motion has an exponential distribution with weight in zero.
reviewed by: Yuliya S. Mishura
description: scientific article; zbMATH DE number 6031480