On stationary and cycle-stationary sequences

Consider random sequences in two-sided time split into cycles by the visits to a recurrent set of states A. The two Palm dualities between stationary sequences Z and sequences with stationary cycles Z° are constructed using the change-of-measure and change-of-origin method. The first duality has the standard interpretation that Z° behaves like Z conditioned on Z₀ ∈ A. The second duality has the less known but no less important interpretation that Z° behaves like Z seen from a typical visit to A.