Variances of first passage times in a Markov chain with applications to mixing times
type: Item
instance of: scholarly article
title: Variances of first passage times in a Markov chain with applications to mixing times
zbMATH Open document ID: 1146.60054
author: Jeffrey J. Hunter
published in: Linear Algebra and its Applications
publication date: 2008-08-06T00:00:00Z
full work available at URL: http://hdl.handle.net/10179/4484
review text: The author [Linear Algebra Appl. 417, No.~1, 108--123 (2006; Zbl 1099.60048)] introduced the quantity \(\eta _{i}=\sum_{j=1}^{m}m_{ij}\pi _{j}\) as a measure of the ``mixing time'' or ``time to stationarity'' in a finite irreducible discrete time Markov chain with state space \(S=\left\{ 1,2,\dots ,m\right\} ,\;m\geq 2\), stationary distribution \((\pi _{j})_{1\leq j\leq m}\), and mean first passage time \(m_{ij}\) from state \(i\) to state \(j\), and showed that \(\eta _{i}:=\eta \) does not actually depend on \(i\in S\). In the present paper, by using generalized inverses of \(I-P\), the author derives expressions for the variances \(v_{i}=\sum_{j=1}^{m}m_{ij}^{\left( 2\right) }\pi _{j}-\eta ^{2},\;i\in S\), where \(m_{ij}^{\left( 2\right) }\) is the second moment of the first passage time from state \(i\) to state \(j\). Here, \(I\) is the unit matrix of order \(m\) and \(P\) the transition matrix of the Markov chain. The special cases \(m=2\) and \(m=3\) are paid special attention.
reviewed by: Marius Iosifescu
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 15B51
Mathematics Subject Classification ID: 60J20
Mathematics Subject Classification ID: 65F20
Mathematics Subject Classification ID: 65F35
zbMATH DE Number: 5307055
zbMATH Keywords: Markov chain
zbMATH Keywords: first passage time
zbMATH Keywords: mixing time
zbMATH Keywords: time to stationarity
MaRDI profile type: Publication
OpenAlex ID: W2162370136
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DOI: 10.1016/J.LAA.2007.06.016
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description: scientific article; zbMATH DE number 5307055