Variances of first passage times in a Markov chain with applications to mixing times (Q935384)

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.