Assume that (X,f) is a dynamical system and that φ is a real non negative potential such that the corresponding f-invariant measure μφ is infinite. Under assumptions of good inducing schemes, we give conditions under which the pressure of f for a perturbed potential φ + sψ relates to the pressure of the induced system. This extends some of Sarig’s results to the setting of infinite ‘equilibrium states’. In addition, limit properties of the family of measures μφ + sψ as s→0 are studied and statistical properties (e.g. correlation coefficients) under the limit measure are derived. I will discuss several examples. This is based on joint work with H. Bruin and M. Todd.