The answer is negative.
It is a consequence of the results of G. Argiris and
J. M. Rosenblatt
[1], obtained in 2001. Also, the answer is given by a recent
note [2] of I. Assani
(2002).
References:
1. G. Argiris and J. M. Rosenblatt, Forcing divergence if the
supremum is not integrable, Preprint (2001)
2. I. Assani, A remark on the martingale theorem and the ergodic
theorem, Preprint (2002)