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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
DOI: https://doi.org/10.1090/S0002-9939-1981-0627697-3
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?)

  • [1] Y. S. Chow, H. Robbins and D. Siegmund, Great expectations: The theory of optimal stopping, Houghton Mifflin, Boston, Mass., 1971. MR 0331675 (48:10007)
  • [2] T. P. Hill and R. P. Kertz, Ratio comparisons of supremum and stop rule expectations, Z. Wahrsch. Verw. Gebiete (to appear). MR 618276 (82h:60078)
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DOI: https://doi.org/10.1090/S0002-9939-1981-0627697-3
Article copyright: © Copyright 1981 American Mathematical Society

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