To be precise, what you're dealing with is rather a moment generating function (kind of Laplace transform) than a characteristic function (kind of Fourier transform).

By the way, it is

*not easy* to prove that a function is a moment generating function for some probability distribution (it involves Bochner's theorem, which is uneasy to check in general), so that you should prove first that your function is indeed a m.g.f.. Actually it

*is* a m.g.function, because this is that of a Gaussian vector...

Marginals are indeed standard Gaussian r.v., but the m.g.f. of the sum is

(take

), which is the m.g.f. of a centered Gaussian with variance 4... Hence this is no counterexample.