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

TL;DR

This paper investigates how well Gaussian distributions can be approximated by truncated distributions and Gauss-Hermite quadrature, focusing on the rates of approximation in the complex Laplace transform domain.

Contribution

It reveals the limitations of truncation for approximating Gaussian Laplace transforms and demonstrates the optimality of Gauss-Hermite quadrature for this purpose.

Findings

Truncation achieves $e^{- heta(a^2)}$ rate in $L_ty$-distance of characteristic functions.

In the complex disk, the optimal approximation rate is $e^{- heta(a^2 \u03bb a)}$, which truncation does not attain.

Gauss-Hermite quadrature attains the optimal approximation rate.

Abstract

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 $e^{-\Theta(a^2)}$. The optimal rate can be attained by the Gauss-Hermite quadrature. As corollary, we also construct a ``super-flat'' Gaussian mixture of $\Theta(a^2)$ components with means in $[-a,a]$ and whose density has all derivatives bounded by $e^{-\Omega(a^2 \log(a))}$ in the $O(1)$-neighborhood of the origin.