We study limit processes for consecutive return-times of asymptotically rare events in general ergodic probability preserving systems. It is shown that in every ergodic system on a non-atomic space any non-negative stationary sequence with expectation not exceeding 1 occurs as a limit process. Moreover, the limiting behaviour is shown to be robust under small changes of the sets. We also determine the relation between asymptotic return-time processes and asymptotic hitting-time processes, and record some consequences.
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