As in traditional probability theory, one of the most difficult problems in the theory of phase transitions concerns the limit distributions for sums of a large number of random variables. However, these variables are strongly dependent. Therefore the usual methods cannot be applied. The limit distributions which appear in these problems are invariant under a subgroup of linear endomorphisms, called the renormalization group. In this paper, we find Gaussian invariant distributions and construct formal series for non-Gaussian ones. Our approach is inspired by the famous renormalization group method widely known in physical literature and developed mainly by K. Wilson, M. Fisher and L. Kadanoff.