Properties of a word-valued source with a non-prefix-free word set

Recently, a word-valued source has been proposed as a new class of information source models. A word-valued source is regarded as a source with a probability distribution over a word set. Although a word-valued source is a nonstationary source in general, it has been proved that an entropy rate of the source exists and the Asymptotic Equipartition Property (AEP) holds when the word set of the source is prefix-free. However, when the word set is not prefix-free (non-prefix-free), only an upper bound on the entropy density rate for an i.i.d. word-valued source has been derived so far. In this paper, we newly derive a lower bound on the entropy density rate for an i.i.d. word-valued source with a finite non-prefix-free word set. Then some numerical examples are given in order to investigate the behavior of the bounds.