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 empirical 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 inequality of Talagrand, we obtain the necessary and sufficient conditions for the large deviation estimations, the moderate deviation estimations and the LIL of L(n)(.) in the Banach space of bounded functionals l(infinity)(F). The extension to the unbounded functionals is also discussed.