Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T20:59:10.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Sequential selection of random vectors under a sum constraint

Part of: Stochastic processes Sequential methods

Published online by Cambridge University Press:  14 July 2016

Mario Stanke
Mario Stanke*
Affiliation:
Georg-August-Universität Göttingen
*
∗ Postal address: Institut für Mikrobiologie und Genetik, Abteilung Bioinformatik, Georg-August-Universität Göttingen, Goldschmidtstraße 1, 37077 Göttingen, Germany. Email address: mstanke@gwdg.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We observe a sequence X 1, X 2,…, X n of independent and identically distributed coordinatewise nonnegative d-dimensional random vectors. When a vector is observed it can either be selected or rejected but once made this decision is final. In each coordinate the sum of the selected vectors must not exceed a given constant. The problem is to find a selection policy that maximizes the expected number of selected vectors. For a general absolutely continuous distribution of the X i we determine the maximal expected number of selected vectors asymptotically and give a selection policy which asymptotically achieves optimality. This problem raises a question closely related to the following problem. Given an absolutely continuous measure μ on Q = [0,1] d and a τ ∈ Q, find a set A of maximal measure μ(A) among all A ⊂ Q whose center of gravity lies below τ in all coordinates. We will show that a simplicial section { x ∈ Q | 〈 x , θ 〉 ≤ 1}, where θ ∈ ℝ d , θ ≥ 0, satisfies a certain additional property, is a solution to this problem.

Keywords

Online selectionsum constraintthreshold region

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 62L15: Optimal stopping
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 1 , March 2004 , pp. 131 - 146
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Coffman, E. G. Jr, Flatto, L., and Weber, R. R. (1987). Optimal selection of stochastic intervals under a sum constraint. Adv. Appl. Prob. 19, 454–473.Google Scholar
Garey, M. R., Graham, R. L., and Johnson, D. S. (1976). Resource constrained scheduling as generalized bin packing. J. Combinatorial Theory A 21, 257–298.Google Scholar
Gnedin, A. V. (2002). Sequential selection under a sum constraint when the number of observations is random. In Limit Theorems in Probabilty and Statistics (Balatonlelle, 1999), Vol. 2, eds Berkes, I., et al., János Bolyai Mathematical Society, Budapest, pp. 91–105.Google Scholar
Luenberger, D. G. (1969). Optimization by Vector Space Methods. John Wiley, New York.Google Scholar
Mallows, C. L., Nair, V. N., Shepp, L. A., and Vardi, Y. (1985). Optimal sequential selection of secretaries. Math. Operat. Res. 10, 709–715.CrossRefGoogle 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.Google Scholar