Lower bounds on the approximation of the multivariate empirical process

Abstract

It is well-known that if C is the class of rectangles 0≦x1≦a1, 0≦x2≦a2 or the class of circular discs then the normalized empirical measure on C behaves like a Brownian bridge. Our main result shows that for these two classes the distances between the normalized empirical measure and the nearest Brownian measure have entirely different order of magnitudes. © 1985 Springer-Verlag.