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Embedding optimal selection problems in a Poisson process. (English) Zbl 0745.60040
A version of the classical “secretary problem” is considered, where the number $$N$$ of candidates available is a random variable. The $$N$$ candidates arrive at times which are independent and uniformly distributed on $$(0,1)$$, and the objective is to minimize a loss which is a non-decreasing function of the ranks of the candidates. This problem has been variously studied by J. Gianini and S. M. Samuels [Ann. Probab. 4, 418-432 (1976; Zbl 0341.60033)], by R. Cowan and J. Zabczyk [Theory Probab. Appl. 23, 584-592 (1979); reprinted from Teor. Veroyatn. Primen. 23, 606-614 (1978; Zbl 0396.62063)], by W. J. Stewart [Applied probability – computer science: the interface, Proc. Meet., Boca Raton/FL 1981, Vol. 1, Prog. Comput. Sci. 2, 275-296 (1982; Zbl 0642.60075)], and by F. T. Bruss [Ann. Probab. 12, 882- 889 (1984; Zbl 0553.60047)] and F. T. Bruss and S. M. Samuels [ibid. 15, 824-830 (1987; Zbl 0592.60034)].
The main contribution of this paper is to show that by imbedding the process in a Poisson process it is possible to obtain all the distributional results necessary to obtain the optimal policy. The special case where $$N$$ is geometrically distributed is particularly simple, and the optimal policy can be found explicitly, but even in the case where $$N$$ has an arbitrary distribution, it is shown that routine calculus methods can be used to prove that the optimal policy is of a certain conjectured form.

##### MSC:
 60G40 Stopping times; optimal stopping problems; gambling theory
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##### References:
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