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

At a positive-rate state the jump chain moves to either other state with probability $1/2$, independently of the holding rate. At a zero-rate state there is no subsequent jump; extend the chain as permanently absorbed there, giving an identity row. This convention is needed to speak of all powers of its transition matrix in the absorbing cases.

When all rates are positive,Its unique equilibrium is uniform and its powers tend to the matrix with all rows $(1/3,1/3,1/3)$. This differs from the holding-time-weighted equilibrium of the continuous-time chain unless the rates coincide.

With one zero-rate state the unique jump-chain equilibrium is its point mass and the limit has all rows that point mass. With two zero-rate states the equilibria are all mixtures on them, and the limiting rows are exactly the absorbing rows and half-half transient row found in part (d).