Conditional large deviations for a sequence of words

Abstract

Cut an i.i.d. sequence ( X i ) of ‘letters’ into ‘words’ according to an independent renewal process. Then one obtains an i.i.d. sequence of words, and thus the level 3 large deviation behaviour of this sequence of words is governed by the specific relative entropy. We consider the corresponding problem for the conditional empirical process of words, where one conditions on a typical underlying ( X i ) . We find that if the tails of the word lengths decay exponentially, the large deviations under the conditional distribution are almost surely again governed by the specific relative entropy, but the set of attainable limits is restricted. We indicate potential applications of such a conditional LDP to the computation of the quenched free energy for directed polymer models with random disorder.