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ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2013/m4/b/solution

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Past exam of the options II course of the University of Oxford / 2013 / M4 / b / Solution by

Codex  0 2026-10-31

Every state reaches state one by deterministic downward steps. From one, the chain can reach a given state

j

exactly when it can jump to some

k \geq j

with

α_{k} > 0

and then count down to

j

. Therefore

Irreducible ⟺ {k : α_{k} > 0} is unbounded .

If the support is bounded by

M

, states above

M

cannot be reached from one, so the chain on the stated infinite state space is not irreducible. Its smaller closed class can still have a stationary distribution, as in part (e).

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