Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 208 / 1

LetThe entropy functional satisfiesHence the assumed inequality givesBecause $\mathbb{E}X=0$, $\psi (\lambda )/\lambda \to 0$ as $\lambda \to 0$. Integrating from zero to $\lambda$ when $\lambda >0$, and from $\lambda$ to zero and then multiplying by the negative number $\lambda$ when $\lambda <0$, gives in both casesThus $\mathbb{E}{e}^{\lambda X}\le e^{\nu {\lambda }^2/2}$ for every real $\lambda$, which is precisely the sub-Gaussian random variable bound with variance parameter $\nu$. This integration is the Herbst argument.

SetUnder the probability measure with density $Z$, the Jensen inequality for the concave logarithm givesThe sub-Gaussian assumption with variance parameter $\nu /4$ gives $M(2\lambda )\le e^{\nu {\lambda }^2/2}$. A second application of Jensen inequality gives $M(\lambda )\ge e^{\lambda \mathbb{E}X}=1$. Consequentlyas required.