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.