In infinite ergodic theory, distributional limits replace
the absolutely normalized pointwise ergodic theorem.
I'll give a review of the subject and then show that every random
variable on the positive reals occurs as the
distributional limit of some infinite ergodic transformation.
This is a consequence of the dual result that every random variable on
the positive reals occurs as the distributional limit ofthe partial sums some positive, ergodic stationary process normalized by a
1-regularly varying normalizing sequence (& the process can be chosen
over any EPPT).
Joint work in progress with Benjamin Weiss.