A stationary process (X_n) is strongly stationary if for every 
positive integer k the process defined by (X_kn) has the same distribution.
This notion was defined by H. Furstenberg and Y. Katznelson who showed its usefulness in establishing extensions of Szemeredi's theorem.    
E. Jenvey proved that any ergodic strongly stationary process is in fact
an independent process.  I will sketch a proof of Jenvey's theorem under the additional assumption that the process is strongly ergodic.