Originally Posted by

**Laurent** 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 $\displaystyle \mathbb{E}[e^{s(X+Y)}]=\exp(2s^2)$ (take $\displaystyle s_1=s_2=s$), which is the m.g.f. of a centered Gaussian with variance 4... Hence this is no counterexample.