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

Authors: T. P. Hill and Robert P. Kertz
Journal: Proc. Amer. Math. Soc. 83 (1981), 582-585
MSC: Primary 60G40
MathSciNet review: 627697
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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.

References [Enhancements On Off] (What's this?)

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Article copyright: © Copyright 1981 American Mathematical Society

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