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Difference prophet inequalities for \([0,1]\)-valued i.i.d. random variables with cost for observations. (English) Zbl 1078.60030

The subject of this article is inequalities in optimal stopping theory [cf. Y. S. Chow, H. Robbins and D. Siegmund, “Great expectations: The theory of optimal stopping” (1971; Zbl 0233.60044)]. More specifically, a comparison of an optimal stopping value and the expected value of the maximum for some sequence of random variables which is called prophet inequalities. The case of independent \([0,1]\)-random variables has been considered by U. Krengel and L. Sucheston [in: Probability on Banach spaces. Adv. Probab. Relat. Top. 4, 197–266 (1978; Zbl 0394.60002); cf. also T. P. Hill and R. P. Kertz [in: Strategies for sequential search and selection in real time. Contemp. Math. 125, 191–207 (1992; Zbl 0794.60040)], and F. Harten, A. Meyerthole and N. Schmitz [“Prophetentheorie. Prophetenungleichungen, Prophetenregionen, Spiele gegen einen Propheten” (1997; Zbl 0886.60033)].
In the present paper the investigation of the difference between the expected gains of the gambler and of the prophet in the case of \([0,1]\)-valued i.i.d. random variables, when there is an observation cost, is given [cf. M. Jones, J. Multivariate Anal. 34, No. 2, 238–253 (1990; Zbl 0753.60042)]. Improvements of the result by E. Samuel-Cahn [Ann. Probab. 20, No. 3, 1222–1228 (1992; Zbl 0777.60035)] are obtained.

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
60E15 Inequalities; stochastic orderings
62L15 Optimal stopping in statistics
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References:

[1] Chow, Y. S., Robbins, H. and Siegmund, D. (1971). Great Expectations : The Theory of Optimal Stopping. Houghton Mifflin, Boston. · Zbl 0233.60044
[2] Harten, F. (1996). Prophetenregionen bei zeitlichen Bewertungen im unabhängigen und im iid-Fall. Ph.D. thesis, Univ. Münster. · Zbl 0855.60042
[3] Harten, F., Meyerthole, A. and Schmitz, N. (1997). Prophetentheorie. Teubner, Stuttgart. · Zbl 0886.60033
[4] Hill, T. P. and Kertz, R. P. (1982). Comparisons of stop rule and supremum expectations of i.i.d. random variables. Ann. Probab. 10 336–345. JSTOR: · Zbl 0483.60035
[5] Jones, M. L. (1990). Prophet inequalities for cost of observation stopping problems. J. Multivariate Anal. 34 238–253. · Zbl 0753.60042
[6] Saint-Mont, U. (1999). Prophet regions for iid random variables with simultaneous costs and discountings. Statist. Decisions 17 185–203. · Zbl 0949.60059
[7] Samuel-Cahn, E. (1992). A difference prophet inequality for bounded i.i.d. variables, with cost for observations. Ann. Probab. 20 1222–1228. JSTOR: · Zbl 0777.60035
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