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.