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

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- Solution a

Solution

Expand according to the hitting time and use detailed balance to reverse each finite path:

P_{π} (X_{T_{A}} = z) = n \geq 0 \sum x_{0}, \dots, x_{n - 1} \in / A \sum π (x_{0}) P (x_{0}, x_{1}) \dots P (x_{n - 1}, z) = π (z) n \geq 0 \sum P_{z} (X_{1}, \dots, X_{n} \in / A) .

The event in the final probability is

{T_{A}^{+} > n}

. The tail-sum formula for a nonnegative integer-valued random variable gives

\sum_{n \geq 0} P_{z} (T_{A}^{+} > n) = E_{z} T_{A}^{+} .

Therefore

P_{π} (X_{T_{A}} = z) = π (z) E_{z} T_{A}^{+} .

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