Summary
Suppose X 1,X 2,...,Xn are independent non-negative random variables with finite positive expectations. Let T n denote the stop rules for X 1,...,X n. The main result of this paper is that E(max{X 1,...,X n }) <2 sup{EX t :tεT n }. The proof given is constructive, and sharpens the corresponding weak inequalities of Krengel and Sucheston and of Garling.
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Chow, Y.S., Robbins, H., Siegmund, D.: Great Expectations: The Theory of Optimal Stopping. Boston: Houghton Mifflin 1971
Edgar, G.A., Sucheston, L.: Amarts; A Class of Asymptotic Martingales. A. Discrete Parameter. J. Multivariate Analysis, 3, 193–221 (1976)
Krengel, U., Sucheston, L.: Semiamarts and Finite Values. Bulletin of the Amer. Math. Soc. 83, 745–747 (1977)
Krengel, U., Sucheston, L.: On Semiamarts, Amarts, and Processes with Finite Value in Probability on Banach Spaces. New York: Marcel Dekker 1978
Krengel, U., Sucheston, L.: How to Bet Against a Prophet. (Some L 1 Dominated Estimates for Semiamarts). Abstract, Notices of the Amer. Math. Soc. 24, A-159 (1977)
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Partially supported by AFOSR Grant F49620-79-C-0123
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Hill, T.P., Kertz, R.P. Ratio comparisons of supremum and stop rule expectations. Z. Wahrscheinlichkeitstheorie verw Gebiete 56, 283–285 (1981). https://doi.org/10.1007/BF00535745
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DOI: https://doi.org/10.1007/BF00535745