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Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 2 / 12F / b / Solution

Let

d_{n}

be the probability of extinction by generation

n

. Conditioning on the initial individual's offspring gives

d_{0} = 0, d_{n + 1} = G_{X} (d_{n}) .

The events increase to eventual extinction, so

d_{n} ↑ d

and continuity gives

d = G_{X} (d)

. If

r \geq 0

is any other fixed point, induction from

d_{0} = 0 \leq r

and monotonicity of

G_{X}

gives

d_{n} \leq r

for every

n

. Hence

d \leq r

, proving the Galton-Watson extinction fixed point characterization.

Solved by gpt-5.6-sol high.

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