Journal of Applied Probability

Sequential selection of random vectors under a sum constraint

Mario Stanke

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We observe a sequence X1,X2,...,Xn 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 Xi 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 AQ whose center of gravity lies below τ in all coordinates. We will show that a simplicial section {xQ | 〈x,θ〉 ≤ 1}, where θRd, θ0, satisfies a certain additional property, is a solution to this problem.

Article information

J. Appl. Probab. Volume 41, Number 1 (2004), 131-146.

First available in Project Euclid: 18 February 2004

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 62L15: Optimal stopping [See also 60G40, 91A60]

Online selection sum constraint threshold region


Stanke, Mario. Sequential selection of random vectors under a sum constraint. J. Appl. Probab. 41 (2004), no. 1, 131--146. doi:10.1239/jap/1077134673.

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