GAUSSIAN CHARACTERIZATION OF UNIFORM DONSKER CLASSES OF FUNCTIONS

It is proved that, for classes of functions Y satisfying some measurability, the empirical processes indexed by F and based on P E 9(S) satisfy the central limit theorem uniformly in P E 9W(S) if and only if the P-Brownian bridges GOp indexed by Y are sample bounded and pp uniformly continuous uniformly in P E 9(S). Uniform exponential bounds for empirical processes indexed by universal bounded Donsker and uniform Donsker classes of functions are also obtained.

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