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A prophet inequality for independent random variables with finite variances

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

D. P. Kennedy  and
R. P. Kertz
D. P. Kennedy*
Affiliation:
University of Cambridge
R. P. Kertz*
Affiliation:
Georgia Institute of Technology
*
∗Postal address: Statistical Laboratory, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, UK.
∗∗Postal address: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA.
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Abstract

It is demonstrated that for each n ≧ 2 there exists a minimal universal constant, cn, such that, for any sequence of independent random variables {Xr, r ≧ 1} with finite variances, , where the supremum is over all stopping times Τ, 1 ≦ Τ ≦ n. Furthermore, cn ≦ 1/2 and lim infn→ ∞cn ≧ 0.439485 · ··.

Keywords

OPTIMAL STOPPINGPROPHET INEQUALITY

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 34 , Issue 4 , December 1997 , pp. 945 - 958
Copyright
Copyright © Applied Probability Trust 1997 

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Footnotes

Supported in part by NSF grant DMS 92-09586.

References

Assaf, D. and Samuel-Cahn, E. (1996) The secretary problem: minimizing the expected rank, with i.i.d random variables. Adv. Appl. Prob. 28, 828–852.Google Scholar
Chung, K. L. (1974) A Course in Probability Theory. 2nd edn. Academic Press, New York.Google Scholar
Dubins, L. E. and Gilat, D. (1978) On the distribution of maxima of martingales. Proc. Amer. Math. Soc. 68, 337–338.Google Scholar
Dubins, L.E. and Pitman, J. (1980) A maximal inequality for skew fields. Z. Wahrscheinlichkeitsth. 52, 219–227.Google Scholar
Dubins, L. E. and Schwarz, G. (1988) A sharp inequality for sub-martingales and stopping times. Soc. Math. de France, Astérisque 157–158, 129–145.Google Scholar
Heijnen, B. and Goovaerts, M. J. (1989) Best upper bounds on risks altered by deductibles under incomplete information. Scand. Act. J. 23–46.Google Scholar
Hill, T. P. and Kennedy, D. P. (1992) Sharp inequalities for optimal stopping with rewards based on ranks. Ann. Appl. Prob. 2, 503–517.CrossRefGoogle Scholar
Hill, T. P. and Kertz, R. P. (1981) Additive comparisons of stop rule and supremum expectations of uniformly bounded independent random variables. Proc. Amer. Math. Soc. 83, 582–585.CrossRefGoogle Scholar
Hill, T. P. and Kertz, R. P. (1983) Stop rule inequalities for uniformly bounded sequences of random variables. Trans. Amer. Math. Soc. 278, 197–207.Google Scholar
Hill, T. P. and Kertz, R. P. (1992) A survey of prophet inequalities in optimal stopping theory. Contemp. Math. 125, 191–207.Google Scholar
Kennedy, D. P. and Kertz, R. P. (1991) The asymptotic behavior of the reward sequence in the optimal stopping of i.i.d. random variables. Ann. Prob. 19, 329–341.Google Scholar
Krengel, U. and Sucheston, L. (1978) On semiamarts, amarts, and processes with finite value. In Probability on Banach Spaces. ed. Kuelbs, J. Marcel Dekker, New York.Google Scholar
Marshall, A. W. and Olkin, I. (1979) Inequalities: Theory of Majorization and its Applications. Academic Press, New York.Google Scholar
Pittenger, A. O. (1990) Sharp mean-variance bounds for Jensen-type inequalities. Statist. Prob. Lett. 10, 91–94.Google Scholar
Resnick, S. I. (1987) Extreme Values, Regular Variation, and Point Processes. Springer, New York.CrossRefGoogle Scholar
Serfozo, R. (1982) Functional limit theorems for extreme values of arrays of independent random variables. Ann. Prob. 10, 172–177.CrossRefGoogle Scholar
Stoyan, D. (1973) Bounds for the extrema of the expected value of a convex function of independent random variables. Studia Sci. Math. Hungar. 8, 153–159.Google Scholar
Tong, Y. L. (1990) The Multivariate Normal Distribution. Springer, New York.Google Scholar
Troutman, J.L. (1996) Variational Calculus and Optimal Control. Springer, New York.CrossRefGoogle Scholar