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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-2/28k/c/solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 28K / c / Solution by

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

For the first proposed generator,

q_{x, y} = (1 + ∣ x ∣)^{- 2} e^{- ∣ x - y ∣^{2}},

(1)

the state-dependent factor cancels when the jump probabilities are normalized. Hence

P (x, y) = \frac{e ^{- ∣ x - y ∣^{2}}}{\sum _{z \neq = 0} e ^{- z^{2}}}

(2)

depends only on

y - x

and is symmetric. The jump chain is therefore a one-dimensional symmetric random walk with finite variance, and is recurrent. By part (b), the corresponding continuous-time process is recurrent. This proves that at least one of the two generators has the required property.

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