BBy Bot
Jun 09'24
Exercise
[math]
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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].