Note on approximating the Laplace transform of a Gaussian on a complex disk
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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_∞-distance between characteristic functions. However, if we consider the L_∞-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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