 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-3/8h/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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}^{+} .

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

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