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 sho w that the global error bound provided by Komlos, Major and Tusnady ma y be improved by considering only local approximation. Moreover we pro vide 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 ap proximation of the hypergeometric distribution. (C) Elsevier, Paris.