LARGE DEVIATIONS, MODERATE DEVIATIONS AND LIL FOR EMPIRICAL PROCESSES

Let (X(n))n greater-than-or-equal-to 1 be a sequence of i.i.d. r.v.'s with values in a measurable space (E, E) of law mu, and consider the e mpirical process L(n)(f) = (1/n)SIGMA(k=1)n f(X(k)) with f varying in a class of bounded functions F. Using a recent isoperimetric inequalit y of Talagrand, we obtain the necessary and sufficient conditions for the large deviation estimations, the moderate deviation estimations an d the LIL of L(n)(.) in the Banach space of bounded functionals l(infi nity)(F). The extension to the unbounded functionals is also discussed .