Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 1 / 27I / d / Solution

When all rates are positive, finite irreducible convergence gives $P(t)\to 1\pi$, with $\pi$ as above. With one zero-rate state $z$, each visit to either transient state has probability $1/2$ of entering $z$ on the next jump. Survival for $n$ jumps therefore has probability $2^{-n}$, so absorption is almost sure andWith two zero-rate states $z_1,z_2$ and the sole positive-rate state $k$, absorption occurs at the first jump, with probability $1/2$ into each absorbing state. The limit rows are $e_{z_1}^T,e_{z_2}^T$ for those states and $(e_{z_1}^T+e_{z_2}^T)/2$ for row $k$. These limits depend on the initial state, as nonuniqueness requires.