## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 3, Number 3 (1993), 794-801.

### A Matching Problem and Subadditive Euclidean Functionals

#### Abstract

A classical paper by Steele establishes a limit theorem for a wide class of random processes that arise in problems of geometric probability. We propose a different (and arguably more general) set of conditions under which complete convergence holds. As an application of our framework, we prove complete convergence of $M(X_1, \ldots, X_n)/\sqrt n$, where $M(X_1, \ldots, X_n)$ denotes the shortest sum of the lengths of $\lfloor n/2\rfloor$ segments that match $\lfloor n/2\rfloor$ disjoint pairs of points among $X_1, \ldots, X_n$, where the random variables $X_1, \ldots, X_n, \ldots$ are independent and uniformly distributed in the unit square.

#### Article information

**Source**

Ann. Appl. Probab., Volume 3, Number 3 (1993), 794-801.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005364

**Digital Object Identifier**

doi:10.1214/aoap/1177005364

**Mathematical Reviews number (MathSciNet)**

MR1233626

**Zentralblatt MATH identifier**

0784.60020

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 60G17: Sample path properties

**Keywords**

Matching problem subadditive functionals complete convergence

#### Citation

Rhee, WanSoo T. A Matching Problem and Subadditive Euclidean Functionals. Ann. Appl. Probab. 3 (1993), no. 3, 794--801. doi:10.1214/aoap/1177005364. https://projecteuclid.org/euclid.aoap/1177005364