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Sequential selection of an increasing subsequence from a sample of random size

Part of: Stochastic processes Sequential methods

Published online by Cambridge University Press:  14 July 2016

Alexander V. Gnedin
Alexander V. Gnedin*
Affiliation:
University of Göttingen
*
∗Postal address: Institute of Mathematical Stochastics, University of Göttingen, Lotzestrasse 13, 37083 Göttingen, Germany. Email address: gnedin@math.uni-goettingen.de.
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Abstract

A random number of independent identically distributed random variables is inspected in strict succession. As a variable is inspected, it can either be selected or rejected and this decision becomes final at once. The selected sequence must increase. The problem is to maximize the expected length of the selected sequence.

We demonstrate decision policies which approach optimality when the number of observations becomes in a sense large and show that the maximum expected length is close to an easily computable value.

Keywords

Sequential selectionincreasing sequencesquare-root law

MSC classification

Primary: 62L15: Optimal stopping 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 1074 - 1085
Copyright
Copyright © Applied Probability Trust 1999 

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References

Baryshnikov, Yu. M., and Gnedin, A. V. (1999). Sequential selection of an increasing sequence from a multidimensional random sample. To appear in Ann. Appl. Prob.Google Scholar
Boshuizen, F. A., and Kertz, R. P. (1999). Smallest-fit selection of random sizes under a sum constraint: weak convergence and moment comparisons. Adv. Appl. Prob. 31, 178–198.CrossRefGoogle Scholar
Bruss, F. T., and Robertson, J. B. (1991). ‘Wald's lemma’ for sums of order statistics of i.i.d. random variables. Adv. Appl. Prob. 23, 612–623.CrossRefGoogle Scholar
Coffman, E. G., Flatto, L., and Weber, R. R. (1987). Optimal selection of stochastic intervals under a sum constraint. Adv. Appl. Prob. 19, 454–473.CrossRefGoogle Scholar
Deuschel, J.-D., and Zeitouni, O. (1995). Limiting curves for i.i.d. records. Ann. Prob. 23, 852–878.CrossRefGoogle Scholar
Gnedin, A. V. (1999). Sequential selection of an increasing sequence from a random sample with geometric sample-size. Preprint.Google Scholar
Gnedin, A. V. (1999). A note on sequential selection from permutations. To appear in Combinatorics, Probability and Computing.Google Scholar
Rhee, W., and Talagrand, M. (1991). A note on the selection of random variables under a sum constraint. J. Appl. Prob. 28, 919–923.CrossRefGoogle Scholar
Samuels, S. M., and Steele, J. M. (1981). Optimal sequential selection of a monotone sequence from a random sample. Ann. Prob. 9, 937–947.CrossRefGoogle Scholar
Steele, J. M. (1995). Variations on the monotone subsequence theme of Erdós and Szekeres. In Discrete Problems and Algorithms, ed. Aldous, D. et al. Springer, New York.Google Scholar