A renormalization group model with non-Gaussian fixed point

We apply short distance scaling to the Wick square of a massive free time zero field and show that the characteristic functionals of the suitably renormalized fields have a short distance limit. The properties of the limiting characteristic functionals allow us to find a class of the other renormalization group invariant processes. They are all non-Gaussian, but can be expressed by superposition of the Gaussians. We also discuss the test function spaces and the pointwise limit of the n-point functions.