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Comparison of optimal value and constrained maxima expectations for independent random variables

Published online by Cambridge University Press:  01 July 2016

Robert P. Kertz
Robert P. Kertz*
Affiliation:
Georgia Institute of Technology
*
∗Postal address: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA.
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Abstract

For all uniformly bounded sequences of independent random variables X 1, X2, ···, a complete comparison is made between the optimal value V(X 1 , X 2, ···) = sup {EXt:t is an (a.e.) finite stop rule for X 1,X 2, ···} and , where Mi (X 1,X 2, ···) is the ith largest order statistic for X 1 , X 2, ··· In particular, for k> 1, the set of ordered pairs {(x, y):x = V(X 1 , X 2, ···) and for some independent random variables X 1 , X 2, ··· taking values in [0, 1]} is precisely the set , where Bk (0) = 0, Bk (1) = 1, and for The result yields sharp, universal inequalities for independent random variables comparing two choice mechanisms, the mortal&s value of the game V(X 1 , X 2, ···) and the prophet&s constrained maxima expectation of the game . Techniques of proof include probability- and convexity-based reductions; calculus-based, multivariate, extremal problem analysis; and limit theorems of Poisson-approximation type. Precise results are also given for finite sequences of independent random variables.

Keywords

SHARP INEQUALITIES FOR INDEPENDENT PROCESSESOPTIMAL STOPPINGPROPHET REGIONSORDER STATISTICSPOISSON APPROXIMATIONEXTREMAL DISTRIBUTIONS

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 2 , June 1986, pp. 311 - 340
Copyright
Copyright © Applied Probability Trust 1986 

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Footnotes

Supported in part by National Science Foundation Grant DMS-84-01604.

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