Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 18H (Markov Chains)

True. If the Markov chain with transition matrix $P$ is irreducible, then for every pair of states $i,j$ there is a pathwith $P_{i_{r-1}{i}_r}>0$ at every step. The support assumption gives $Q_{i_{r-1}{i}_r}>0$ for every edge of the same path. Thus every state can reach every other state under $Q$, so the second chain is also an irreducible Markov chain.

True. For a state $i$, letbe its possible return times under $P$, and define ${\mathcal{R}}_Q(i)$ similarly. Every positive $P$-path is a positive $Q$-path, soThe greatest common divisor of the larger set divides that of the smaller set. Since the latter is one by the assumed aperiodic Markov chain property, the former is also one. Hence every state is aperiodic under $Q$.

False. On the two-state space, takeEvery positive entry of $P$ remains positive in $Q$. Under $P$, both singleton states are closed communicating classes, so neither state is a transient state. Under $Q$, however, state one eventually moves to the absorbing state two and can never return after doing so. State one is therefore transient.

False. Again use two states, but now takeThe support condition holds. Under $P$, the first return time to state one is identically one, so ${\mu }_1=1$. Under $Q$, the chain is irreducible with stationary distribution $(1/2,1/2)$. The mean recurrence time formula givesThus enlarging transition support does not imply a smaller mean return time.