## The Annals of Applied Probability

### A Matching Problem and Subadditive Euclidean Functionals

WanSoo T. Rhee

#### 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

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