Strong approximations of bivariate uniform empirical processes

In 1975, Komlos, Major and Tusnady constructed a strong approximation of the uniform empirical process {alpha(n)(t), n greater than or equal to 1, t is an element of [0, 1]} by a Gaussian Kiefer process. We show that the global error bound provided by Komlos, Major and Tusnady may be improved by considering only local approximation. Moreover we provide explicit constants. We also prove a local refinement for Tusnady's Gaussian strong approximation of the bidimensional uniform empirical process. The main technical tool we use is a non asymptotic normal approximation of the hypergeometric distribution. (C) Elsevier, Paris.