Hungarian Constructions from the Nonasymptotic Viewpoint

Abstract

Let x1,…,xn be independent random variables with uniform distribution over [0,1], defined on a rich enough probability space Ω. Denoting by ?̂ n the empirical distribution function associated with these observations and by αn the empirical Brownian bridge αn(t)=n‾√(?̂ n(t)−t), Komlos, Major and Tusnady (KMT) showed in 1975 that a Brownian bridge ?0 (depending on n) may be constructed on Ω in such a way that the uniform deviation ‖αn−?0‖∞ between αn and ?0 is of order of log(n)/n‾√ in probability. In this paper, we prove that a Poisson bridge ?0n may be constructed on Ω (note that this construction is not the usual one) in such a way that the uniform deviations between any two of the three processes αn,?0n and ?0 are of order of log(n)/n‾√ in probability. Moreover, we give explicit exponential bounds for the error terms, intended for asymptotic as well as nonasymptotic use.