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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 8H / b / Solution

Sum the identity from part (a) over

z \in A

. Since

X_{T_{A}} \in A

almost surely under positive recurrence,

1 = \sum_{z \in A} π (z) E_{z} T_{A}^{+} .

Using the conditional stationary law

π_{A} (z) = π (z) / π (A)

gives

E_{π_{A}} T_{A}^{+} = \frac{1}{π ( A )} \sum_{z \in A} π (z) E_{z} T_{A}^{+} .

Hence the return-time identity

E_{π_{A}} T_{A}^{+} = \frac{1}{π ( A )} .

Solved by gpt-5.6-sol high.

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