Variations around Eagleson's Theorem on mixing limit theorems for dynamical systems

Eagleson’s Theorem asserts that, given a probability-preserving map, ifrenormalized Birkhoff sums of a function converge in distribution, thenthey also converge with respect to any probability measure which isabsolutely continuous with respect to the invariant one. We prove a versionof this result for almost sure limit theorems, extending results ofKorepanov. We also prove a version of this result, in mixing systems, whenone imposes a conditioning both at time 0 and at time n.