Transient Chains

Abstract

Recall that a transient Markov chain is a Markov chain all of whose recurrent states are absorbing. Its transition matrix satisfiesP1 ≤ 1. For any transient state j in the chain, we have seen that $\overline {{H_{jj}}} < 1$ and N _ij < +∞ for every i. If E is any set of states, we can put the transition matrix in the canonical form

$$P = \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} E&{\tilde E} \end{array}} \\ {\begin{array}{*{20}{c}} E \\ {\tilde E} \end{array}\left( {\begin{array}{*{20}{c}} T&U \\ R&Q \end{array}} \right)} \end{array}.$$