## Journal of Applied Probability

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

### Sequential selection of random vectors under a sum constraint

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 18 February 2004

**Permanent link to this document**

http://projecteuclid.org/euclid.jap/1077134673

**Digital Object Identifier**

doi:10.1239/jap/1077134673

**Mathematical Reviews number (MathSciNet)**

MR2036277

**Zentralblatt MATH identifier**

1054.60054

**Subjects**

Primary: 60G50: Sums of independent random variables; random walks

Secondary: 62L15: Optimal stopping [See also 60G40, 91A60]

**Keywords**

Online selection sum constraint threshold region

#### Citation

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. http://projecteuclid.org/euclid.jap/1077134673.