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 take

P = (1001), Q = (\frac{1}{2} \frac{1}{2} \frac{1}{2} \frac{1}{2}) .

The support condition holds. Under

P

, the first return time to state one is identically one, so

μ_{1} = 1

. Under

Q

, the chain is irreducible with stationary distribution

(1/2, 1/2)

η_{1} = \frac{1}{π _{1}} = 2 > 1 = μ_{1} .

Thus enlarging transition support does not imply a smaller mean return time.

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