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ABsTsAcr. Sufficient conditions of moment type are given for the existence of optimal stopping rules for Sn/n. These results make use of a generalization of a result of Dini. In general it is shown that necessary conditions of moment type do not exist. In particular, we exhibit a symmetric random variable with EX+(og+ X)+` oo for which an optimal rule exists. However under regularity conditions, we show that the sufficient condition given is very nearly necessary. Some results on exponentials of regularly varying functions are of independent interest.

Moment conditions for the existence and nonexistence of optimal stopping rules for $S\sb{n}/n\sp{1}$