Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 2 / c

Table of contents
- Solution c

Solution

For every

θ \geq 0

, the exponential Markov bound and independence give

P (S_{n} / n \geq x) = P (e^{θ S_{n}} \geq e^{n θ x}) \leq e^{- n θ x} E e^{θ S_{n}} = exp {- n (θ x - ψ (θ))} .

Taking the infimum over

θ \geq 0

yields

lim sup_{n \to \infty} \frac{1}{n} lo g P (S_{n} / n \geq x) \leq - sup_{θ \geq 0} (θ x - ψ (θ)) .

For

x \geq m

, convexity makes this supremum equal to

ψ^{*} (x)

, proving the upper bound in the stated tail form of Cramér theorem.

Solved by gpt-5.6-sol high.

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