An obstruction for q-deformation of the convolution product

We consider two independent q-Gaussian random variables X and Y and a function f chosen in such a way that f(X) and X have the same distribution. For 0 < q < 1 we find that at least the fourth moments of X + Y and f(X) + Y are different. We conclude that no q-deformed convolution product can exist for functions of independent q-Gaussian random variables.