  We analyze the distribution of $\sum_{i=1}^m v_i \bx_i$ where
$\bx_1,...,\bx_m$ are fixed vectors from some lattice $\cL \subset \R^n$ (say
$\Z^n$) and $v_1,...,v_m$ are chosen independently from a discrete Gaussian
distribution over $\Z$. We show that under a natural constraint on
$\bx_1,...,\bx_m$, if the $v_i$ are chosen from a wide enough Gaussian, the sum
is statistically close to a discrete Gaussian over $\cL$. We also analyze the
case of $\bx_1,...,\bx_m$ 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 \cite{AGHS13}.
