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I am currently learning about lattice-based cryptography and, reading from A Decade of Lattice Cryptography by Peikert, specifically section 2.3, it emerges that

[...] if the parameter s is greater or equal than the smoothing parameter of a lattice, then the sum of independent discrete gaussians (over that lattice) is a discrete gaussian itself.

I am looking for the formal statement (and proof) of that fact without any success. Is anyone able to point me to the appropriate reference?

EDIT: added link to the paper

Jackmill
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You mention

Unfortunately each reference is very dense and carries a lot of highly non-trivial results. I'm sure that an exhaustive answer is to be found in one or more of the references, but from a first scan of them I did not find a clear statement of the result.

This can be found in Theorem 4.6 of Improved Discrete Gaussian and Subgaussian Analysis for Lattice Cryptography.

Mark Schultz-Wu
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