[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

An ergodic Markov chain is started in equilibrium

(i.e., with initial probability vector [math]\mat{w}[/math]). The mean time until the next occurrence of state [math]s_i[/math] is [math]\bar{m_i} = \sum_k w_k m_{ki} + w_i r_i[/math]. Show that [math]\bar {m_i} = z_{ii}/w_i[/math], by using the facts that [math]\mat {w}\mat {Z} = \mat {w}[/math] and [math]m_{ki} = (z_{ii} - z_{ki})/w_i[/math].