 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ib/paper-4/9c/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Let

T_{i}^{+} = in f {n \geq 1 : X_{n} = i}

be the first positive return time. A recurrent state is one for which

P_{i} (T_{i}^{+} < \infty) = 1.

The subscript means that the Markov chain starts at

i

. The positive-time condition distinguishes a genuine return from the automatic visit at time zero.

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