OurBigBook.comSite Source code

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

The statement is 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)

. The mean recurrence time formula gives

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

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

Solved by gpt-5.6-sol high.

 Ancestors (11)