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Abstract
A complete determination is made of the possible values for $E\left ( {\sup {X_n}} \right )$ and $\sup \left \{ {E{X_t}:t\;{\text {a}}\;{\text {stop rule}}} \right \}$ for ${X_1},{X_2}, \ldots$ independent uniformly bounded random variables; this yields results of Krengel, Sucheston, and Garling, and of Hill and Kertz as easy corollaries. In optimal stopping problems with independent random variables where the player is free to choose the order of observation of these variables it is shown that the player may do just as well with a prespecified fixed ordering as he can with order selections which depend sequentially on past outcomes. A playerâs optimal expected gain if he is free to choose the order of observation is compared to that if he is not; for example, if the random variables are nonnegative and independent, he may never do better than double his optimal expected gain by rearranging the order of observation of a given sequence.
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