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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-26/2/c/solution

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Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 26 / 2 / c / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

The family is exponentially tight at large-deviation speed

a_{L}

if for every

M > 0

there is a compact set

K_{M} \subset E

such that

L lim sup \frac{1}{a _{L}} lo g P (X^{L} \in / K_{M}) \leq - M .

(1)

For this question

a_{L} = L

. The definition says that escape from an appropriate compact set can be made negligible at any prescribed exponential rate. Requiring a strict inequality instead gives the same notion, by choosing a larger

M

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