 Solution

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

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

In a transient irreducible chain, every state

j

is transient, so the expected number of visits satisfies

\sum_{n = 0}^{\infty} P^{n} (i, j) = E_{i} V_{j} < \infty.

The terms are nonnegative, and every convergent series has terms tending to zero. Therefore

P^{n} (i, j) \to 0 .

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