Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 208 / 1 / c / Solution

The Chernoff bound gives, for $0<\lambda <1/c$,Writing $u=ct/\nu$, elementary differentiation shows that the exponent is minimized atSubstitution givesThus $h$ is the Legendre transform generated by the sub-Gamma cumulant bound after its natural rescaling.