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Let X n be a sequence of real-valued random variables adapted to an increasing sequence of a-algebras F w . We denote by T, Tp T respectively the collection of bounded, finite, and arbitrary stopping times for (F w ) nGN . This paper reports on recent progress concerning the theory of semiamarts, i.e. processes for which (EX T \^T is bounded, initiated in [3], and the theory of amarts, Le. processes for which lim rG:r EX r exists. We relate the notion of semiamart to processes of interest in the theory of optimal stopping (cf.[2] ), namely X n such that \EX^\ < °° for M G Tp or for nET. For independent random variables X n and for processes of the form X n = c~* 2^-j Y t with increasing c"'s and independent nonnegative F f .'s, a new dominated estimate £(sup X£) < K sup EXp (=KV(T))with K = 2 in the first and K < 5.46 in the second case, shows that such processes are semiamarts if and only if suplX" I is integrable. Also in the case when F" = F m for all n 9 m E N, a semiamart has a necessarily integrable supremum. This observation is used to construct averages of aperiodic stationary sequences, which are not semiamarts-thereby strengthening a result announced by A. Bellow [1]. This can be done also in the "descending" case, i.e. when the time domain N is replaced by -N (see [3]); thus our results indicate that there are no connections between the amart theory and the ergodic theory of point transformations.

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