关于复圆盘上高斯的拉普拉斯变换逼近的注记
Note on approximating the Laplace transform of a Gaussian on a complex disk
摘要：
在这个简短的说明中，我们研究如何以及高斯分布可以近似分布支持$[-a,a]$。也许，自然的猜想是，对于大的$a$，通过将高斯截断为$[-a,a]$，给出了几乎最优的选择。实际上，这种近似在特征函数之间的$L_\infty$距离方面实现了$e^{-Θ(a^2)}$的最优速率。然而，如果我们考虑一个复杂的磁盘上的拉普拉斯变换之间的$L_\infty$距离，最佳速率是$e^{-Θ(a^2 \log a)}$，而截断仍然只能达到$e^{-Θ(a^2)}$。最佳速率可以通过Gauss-Hermite求积来获得。作为推论，我们还构建了一个“超平坦”的高斯混合的$Θ(a^2)$组件的平均值在$[-a,a]$和其密度的所有衍生物有界的$e^{-Ω(a^2 \log(a))}$在$O(1)$ -邻域的起源。
英文摘要：
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 $e^{-Θ(a^2)}$ 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 $e^{-Θ(a^2 \log a)}$, while truncation still only attains $e^{-Θ(a^2)}$. The optimal rate can be attained by the Gauss-Hermite quadrature. As corollary, we also construct a ``super-flat'' Gaussian mixture of $Θ(a^2)$ components with means in $[-a,a]$ and whose density has all derivatives bounded by $e^{-Ω(a^2 \log(a))}$ in the $O(1)$-neighborhood of the origin.
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