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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0627697-3
PII: S 0002-9939(1981)0627697-3
Article copyright: © Copyright 1981 American Mathematical Society