## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 3 (1981), 365-376.

### Subadditive Euclidean Functionals and Nonlinear Growth in Geometric Probability

#### Abstract

A limit theorem is established for a class of random processes (called here subadditive Euclidean functionals) which arise in problems of geometric probability. Particular examples include the length of shortest path through a random sample, the length of a rectilinear Steiner tree spanned by a sample, and the length of a minimal matching. Also, a uniform convergence theorem is proved which is needed in Karp's probabilistic algorithm for the traveling salesman problem.

#### Article information

**Source**

Ann. Probab. Volume 9, Number 3 (1981), 365-376.

**Dates**

First available: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176994411

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aop/1176994411

**Mathematical Reviews number (MathSciNet)**

MR626571

**Zentralblatt MATH identifier**

0461.60029

**Subjects**

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Secondary: 60F15: Strong theorems 60C05: Combinatorial probability 60G55: Point processes

**Keywords**

Subadditive process traveling salesman problem Steiner tree Euclidean functional

#### Citation

Steele, J. Michael. Subadditive Euclidean Functionals and Nonlinear Growth in Geometric Probability. The Annals of Probability 9 (1981), no. 3, 365--376. doi:10.1214/aop/1176994411. http://projecteuclid.org/euclid.aop/1176994411.