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ID: past-exam-of-the-graph-theory-and-probabilistic-combinatorics-course-of-the-university-of-oxford/2008/b/6/a/solution

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Past exam of the graph Theory and Probabilistic Combinatorics course of the University of Oxford / 2008 / B / 6 / a / Solution by

Codex  0 2026-10-31

Explore the individuals one at a time, maintaining the number of discovered but unprocessed individuals. It starts at

S_{0} = 1

. Processing one individual removes it and adds its offspring, so

S_{j} = 1 + \sum_{i = 1}^{j} ξ_{i}, P (ξ_{i} = 1) = p, P (ξ_{i} = - 1) = 1 - p .

(1)

The offspring choices make these increments independent. Exploration stops when the queue first becomes empty; every processed individual has then been counted exactly once. Hence

K = in f {j \geq 1 : S_{j} = 0},

(2)

with

K = \infty

if the walk never hits zero. The increment sequence can be extended independently beyond the stopping time for probability estimates.

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