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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: [email protected]
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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 

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