The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 4, Number 4 (1994), 1057-1073.
Limit Theorems and Rates of Convergence for Euclidean Functionals
Abstract
A Beardwood-Halton-Hammersley type of limit theorem is established for a broad class of Euclidean functionals which arise in stochastic optimization problems on the $d$-dimensional unit cube. The result, which applies to all functionals having a certain "quasiadditivity" property, involves minimal structural assumptions and holds in the sense of complete convergence. It extends Steele's classic theorem and includes such functionals as the length of the shortest path through a random sample, the minimal length of a tree spanned by a sample, the length of a rectilinear Steiner tree spanned by a sample and the length of a Euclidean matching. A rate of convergence is proved for these functionals.
Article information
Source
Ann. Appl. Probab., Volume 4, Number 4 (1994), 1057-1073.
Dates
First available in Project Euclid: 19 April 2007
Permanent link to this document
https://projecteuclid.org/euclid.aoap/1177004902
Digital Object Identifier
doi:10.1214/aoap/1177004902
Mathematical Reviews number (MathSciNet)
MR1304772
Zentralblatt MATH identifier
0812.60033
JSTOR
links.jstor.org
Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60F15: Strong theorems 60C05: Combinatorial probability
Keywords
Subadditive and superadditive functionals TSP Steiner tree minimal spanning tree minimal matching rates of convergence
Citation
Redmond, C.; Yukich, J. E. Limit Theorems and Rates of Convergence for Euclidean Functionals. Ann. Appl. Probab. 4 (1994), no. 4, 1057--1073. doi:10.1214/aoap/1177004902. https://projecteuclid.org/euclid.aoap/1177004902
