Past exam of the options II course of the University of Oxford / 2013 / M4 / b / Solution

Every state reaches state one by deterministic downward steps. From one, the chain can reach a given state $j$ exactly when it can jump to some $k\ge j$ with ${\alpha }_k>0$ and then count down to $j$. ThereforeIf the support is bounded by $M$, states above $M$ cannot be reached from one, so the chain on the stated infinite state space is not irreducible. Its smaller closed class can still have a stationary distribution, as in part (e).