 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-4/7h/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Let

q = P_{i} (T_{i} < \infty)

. By the Strong Markov property, after every return to

i

the chance of another return is again

q

. Hence

P_{i} (V_{i} = k) = (1 - q) q^{k - 1}, k = 1, 2, \dots .

Thus $V_{i}$ is geometric with success parameter $1 - q$ ; when $q = 1$ , it is infinite almost surely.

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