Abstract: In this talk we consider the taboo counterpart of stationarity. Stationarity is the characterizing property of any two-sided limit process obtained by shifting the time-origin of a one-sided process to the far future. Similarly, taboo stationarity is the characterizing property of any two-sided limit process obtained by shifting the origin of a one-sided process to the far future "under taboo", that is, conditionally on the process not having entered a taboo region of its state space up to the new time-origin. This is, for instance, an appropriate model for a fish population that has lived a long time in an isolated lake, will eventually become extinct, but is still non-extinct at the time of observation.

We present a basic but amazingly simple structural characterization of taboo stationary processes and then take a closer look at the structure in the regenerative case.