## The Annals of Applied Probability

### Asymptotics for the length of a minimal triangulation on a random sample

J. E. Yukich

#### Abstract

Given $F \subset [0, 1]^2$ and finite, let $\sigma(F)$ denote the length of the minimal Steiner triangulation of points in F. By showing that minimal Steiner triangulations fit into the theory of subadditive and superadditive Euclidean functionals, we prove under a mild regularity condition that $$\lim_{n \to \infty} \sigma(X_1,\dots, X_n)/n^{1/2} = \beta \int_{[0, 1]^2}f(x)^{1/2} dx \c.c.,$$ where $X_1,\dots, X_n$ are i.i.d. random variables with values in $[0, 1]^2$, $\beta$ is a positive constant, f is the density of the absolutely continuous part of the law of $X_1$ , and c.c. denotes complete convergence. This extends the work of Steele. The result extends naturally to dimension three and describes the asymptotics for the probabilistic Plateau functional, thus making progress on a question of Beardwood, Halton and Hammersley. Rates of convergence are also found.

#### Article information

Source
Ann. Appl. Probab., Volume 9, Number 1 (1999), 27-45.

Dates
First available in Project Euclid: 21 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1029962596

Digital Object Identifier
doi:10.1214/aoap/1029962596

Mathematical Reviews number (MathSciNet)
MR1682604

Zentralblatt MATH identifier
0931.68046

#### Citation

Yukich, J. E. Asymptotics for the length of a minimal triangulation on a random sample. Ann. Appl. Probab. 9 (1999), no. 1, 27--45. doi:10.1214/aoap/1029962596. https://projecteuclid.org/euclid.aoap/1029962596

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