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

2020/08/31 by Polyanskiy, Yury, Wu, Yihong
#FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2008.13372

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^-Θ(a2) in terms of the L_∞-distance between characteristic functions. However, if we consider the L_∞-distance between Laplace transforms on a complex disk, the optimal rate is e^{-Θ(a2 log a)}, while truncation still only attains e^-Θ(a2). The optimal rate can be attained by the Gauss-Hermite quadrature. As corollary, we also construct a ``super-flat'' Gaussian mixture of Θ(a²) components with means in [-a,a] and whose density has all derivatives bounded by e^{-Ω(a2 log(a))} in the O(1)-neighborhood of the origin.

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