 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ib/paper-1/19h/c/solution

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

In the positive-recurrent case, let

q = p_{0} + p_{1} < p

. Detailed balance for the length chain requires

π_{k} q = π_{k + 1} p,

so, with

r = q / p < 1

π_{k} = π_{0} r^{k} .

Normalization gives

π_{0} = 1 - r

. The self-loop at zero makes the length chain aperiodic, so the convergence theorem for irreducible positive-recurrent aperiodic Markov chains gives

n \to \infty lim P (ℓ (X_{n}) = k ∣ X_{0} = \emptyset) = (1 - \frac{p _{0} + p _{1}}{p}) (\frac{p _{0} + p _{1}}{p})^{k} .

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