Large deviations for the empirical process of a symmetric measure: a lower bound

Let {Xj}j=1[infinity] be a sequence of r.v.'s defined on a probability space ([Omega],F,[mu]) and taking values in a compact metric space S, let Rn([omega],.)=(1/n)[summation operator]k=0n-1 [delta]Tk(X(n,[omega]))(·) with X(n,[omega]) the point in SZ obtained by repeating (X1([omega]),...,Xn([omega])) periodically on both sides and T the shift on SZ, be the empirical process associated to {Xj}j=1[infinity]. We prove here that a large deviations result in the distributions of the empirical process w.r.t. a certain measure [mu]. This gives large deviations for the distributions of the empirical process with respect to a symmetric measure and also those associated to an exchangeable sequence of r.v.'s.