Abstract
A classical result due to Eagleson states (in particular) that if appropriately normalized Birkhoff sums generated by a measurable function and an ergodic probability preserving transformation converge in distribution, then they also converge in distribution with respect to any probability measure which is absolutely continuous with respect to the invariant one. In this note, we prove several quantitative and infinite-dimensional versions of Eagleson's theorem for some classes of non-stationary stochastic processes which satisfy certain type of decay of correlations.
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