Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 201 / 3 / b

For every $\theta \ge 0$, the Markov inequality and independence give the Chernoff boundOptimize over nonnegative $\theta$. For $a\ge \mu$, part (a) identifies this supremum with $I(a)$, yielding the finite-$n$ estimateIf $I(a)=\infty$, use parameters with arbitrarily large $\theta a-\psi (\theta )$; the probability must then be zero. Taking the logarithmic upper limit proves the upper bound in the Cramér theorem:For $a<\mu$, the nonnegative-parameter supremum is zero, giving the correct trivial tail-rate upper bound. Exponential Markov bounds and the product moment-generating function supply the entire upper-bound argument.