  Scale invariance and the resulting power law behaviours are seen in diverse
systems. In this work we consider translation, rotational and scale invariant
systems defined on a lattice, such that the variables defining the state at
every lattice site take on the same range of finite values, with these values
collectively picked up from probability distribution that can be arbitrary. We
show that the exponent that describes the scaling of the two point correlation
function in these systems will match the scaling exponent of a equilibrium
statistical mechanical model described by a Boltzmannian distribution at
criticality. This work therefore extends the concept of universality in
statistical mechanics to probability distributions that do not have a
Boltzmannian form.
