Additive comparisons of stop rule and supremum expectations of uniformly bounded independent random variables

Abstract:

Let ${X_1},{X_2}, \ldots$ be independent random variables taking values in [$[a,b]$], and let $T$ denote the stop rules for ${X_1},{X_2}, \ldots$. Then $E({\sup _{n \geqslant 1}}{X_n}) - \sup \{ E{X_t}:t \in T\} \leqslant (1/4)(b - a)$, and this bound is best possible. Probabilistically, this says that if a prophet (player with complete foresight) makes a side payment of $(b - a)/8$ to a gambler (player using nonanticipating stop rules), the game becomes at least fair for the gambler.