We prove a local central limit theorem for the sum of one-dimensional discrete Gaussians in $n$-dimensional space. In more detail, we analyze the distribution of $sum_{i=1}^m v_i mathbf{x}_i$ where $mathbf{x}_1,ldots,mathbf{x}_m$ are fixed vectors from some lattice $mathcal{L} subset mathbb{R}^n$ and $v_1,ldots,v_m$ are chosen independently from a discrete Gaussian distribution over $mathbb{Z}$. We show that under a natural constraint on $mathbf{x}_1,ldots,mathbf{x}_m$, if the $v_i$ are chosen from a wide enough Gaussian, the sum is statistically close to a discrete Gaussian over $mathcal{L}$. We also analyze the case of $mathbf{x}_1,ldots,mathbf{x}_m$ that are themselves chosen from a discrete Gaussian distribution (and fixed). Our results simplify and qualitatively improve upon a recent result by Agrawal, Gentry, Halevi, and Sahai
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