Positions of the ranks of factors in certain finite long length words

We consider the set of finite random words A⋆, with independent letters drawn from a finite or infinite totally ordered alphabet according to a general probability distribution. On a specific subset of A⋆, we consider certain factorization of the words. The factors of a word are labelled with ranks, based on the lexicographical order. In this paper we prove that the normalized position of the ranks is uniform, when the length of the word goes to infinity.