 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/ia/paper-2/12f/d/solution

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

P (X_{n} = 0) = G_{n} (0) = 1 - α^{(1 - β^{n}) / (1 - β)} ⟶ 1 - α^{1/ (1 - β)} .

To identify this limit with ultimate extinction, let

E_{n} = {X_{n} = 0}

. The absorbing-zero property of the branching process gives

E_{n} \subseteq E_{n + 1}

. Ultimate extinction means that some finite generation is empty, namely

E = ⋃_{n \geq 0} E_{n}

. Continuity of probability on increasing events therefore gives

P (E) = n \to \infty lim P (E_{n}) = 1 - α^{1/ (1 - β)} .

This set argument is essential: it uses absorption, rather than inferring extinction merely from a numerical limit of generation probabilities.

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