Is the sum of i.i.d. discrete Gaussian rv still distributed as a discrete Gaussian?

Question

Let assume I have $n$ i.i.d. random variables $X_i \; i=1,\dots,n$. Each of them is distributed as $X_i \sim \mathcal{D}_{L,\sigma,c}$, namely a discrete Gaussian distribution over a Lattice $L$ of parameters $\sigma,c \in \mathbb{R}$. Any of you have an idea of how to show that $Y = \sum_{i=1}^{n} X_i \alpha_i$ with $\alpha_i \in \mathbb{R}$ is still distributed as a discrete Gaussian, but with different parameters?