 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ib/paper-3/8h/a/solution

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

States

i

and

j

communicate when each is accessible from the other: there exist

m, n \geq 0

with

(P^{m})_{ij} > 0, (P^{n})_{ji} > 0.

This is an equivalence relation, and its equivalence classes are the communicating classes. A class is closed when no positive-probability transition leaves it.

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