On time- and cycle-stationarity

Consider processes split into cycles by a sequence of random times (called points). The standard Palm relationship between stationary processes with cycles and processes with stationary cycles is produced in two transparent steps: length-biasing and re-centring. It has the following standard intuitive interpretation: the process with stationary cycles behaves like the stationary one conditioned on a point at time zero. A less known modification of this relationship is produced by conditioning on the invariant σ-algebra before length-biasing. It has the following intuitive interpretation: the stationary process behaves like the cycle-stationary one centred at a time chosen at random on the line. The present approach leads to strong conditioning, limit and coupling results motivating these interpretations.