Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 18H / d / Solution

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.