A Matching Problem and Subadditive Euclidean Functionals

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(X₁, …, Xₙ)/√ n, where M(X₁, …, Xₙ) denotes the shortest sum of the lengths of n/2 segments that match n/2 disjoint pairs of points among X₁, …, Xₙ, where the random variables X₁, …, Xₙ, … are independent and uniformly distributed in the unit square.