  Generalized universality, as recently proposed, postulates a universal
non-Gaussian form of the probability density function (PDF) of certain global
observables for a wide class of highly correlated systems of finite volume N.
Studying the 2D XY -model, we link its validity to renormalization group
properties. It would be valid if there were a single dimension 0 operator, but
the actual existence of several such operators leads to T-dependent
corrections. The PDF is the Fourier transform of the partition function Z(q) of
an auxiliary theory which differs by a dimension 0 perturbation with a very
small imaginary coefficient iq/N from a theory which is asymptotically free in
the infrared. We compute the PDF from a systematic loop expansion of ln Z(q).
