Abstract
It is well-known that ifC is the class of rectangles 0≦x 1≦a 1, 0≦x 2≦a 2 or the class of circular discs then the normalized empirical measure onC 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.
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