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

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.

References in corpus (1)

• Self-regularizing Property of Nonparametric Maximum Likelihood Estimator in Mixture Models

Cited by in corpus (1)

• Self-regularizing Property of Nonparametric Maximum Likelihood Estimator in Mixture Models