Note on approximating the Laplace transform of a Gaussian on a complex disk

Understand

In this short note we study how well a Gaussian distribution can be approximated by distributions supported on $[-a,a]$.

• Perhaps, the natural conjecture is that for large $a$ the almost optimal choice is given by truncating the Gaussian to $[-a,a]$.
• Indeed, such approximation achieves the optimal rate of $e^{-\Theta(a^2)}$ in terms of the $L_\infty$-distance between characteristic functions.
• However, if we consider the $L_\infty$-distance between Laplace transforms on a complex disk, the optimal rate is $e^{-\Theta(a^2 \log a)}$, while truncation still only attains $e^{-\Theta(a^2)}$.