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

We consider the set of finite random words \(\mathcal A^\star\), with independent letters drawn from a finite or infinite totally ordered alphabet according to a general probability distribution. On a specific subset of \(\mathcal A^\star\), considering certain factorization of the words which are labelled with the ranks, base on the lexicographical order, we prove that the normalized position of the ranks of factors, are uniform, when the length of the word goes to infinity.