Normal fluctuations and the FKG inequalities

In a translation invariant pure phase of a ferromagnet, finite susceptibility and the FKG inequalities together imply convergence of the block spin scaling limit to the infinite temperature Gaussian fixed point. This result is presented in a rather general probabilistic context and is applicable to infinite cluster density fluctuations in percolation models and to boson field fluctuations in (Euclidean) Yukawa quantum field theory models as well as to magnetization fluctuations in Ising models.