Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 208 / 1 / a / Solution

If $X$ is sub-Gaussian with variance parameter $\nu$, thenfor every $\lambda \in \mathbb{R}$. Restricting this inequality to $|\lambda |<1/\alpha$ proves that $X$ is sub-exponential with parameters $(\nu ,\alpha )$ for every $\alpha >0$.

Now let $X=Z^2-1$ for a standard normal distribution variable $Z$. Its moment-generating function isand is infinite for $\lambda \ge 1/2$. A sub-Gaussian moment-generating function must be finite for every real $\lambda$, so $X$ cannot be sub-Gaussian with any finite parameter. For $|\lambda |<1/4$, the stated inequality givesThus $X$ is sub-exponential with parameters $(4,4)$.