Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 353 / 4 / a / iii

Table of contents
- Solution iii

Solution

Assume

μ_{2} > μ_{1}

, so the upper tail is governed by its boundary. The large deviation principle gives the exponentially equivalent estimate

P (N_{t} / t \geq μ_{2}) ≍ e^{- t I (μ_{2})},

where now

I (x) = x lo g (x / μ_{1}) - x + μ_{1}

. Increasing capacity to

μ_{3} > μ_{2}

reduces this estimate by at least

1 0^{6}

when

t [I (μ_{3}) - I (μ_{2})] \geq 6 lo g 10 .

Equivalently,

μ_{3} lo g \frac{μ _{3}}{μ _{1}} - μ_{3} - μ_{2} lo g \frac{μ _{2}}{μ _{1}} + μ_{2} \geq \frac{6 lo g 10}{t} .

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