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)