Sequential densities of rational languages

Abstract

We introduce the notion of density of a rational language with respect to a sequence of probability measures. We prove that if $(\hat{I}{¼}_n)$ is a sequence of Bernoulli measures converging to a positive Bernoulli measure $\overline{\hat{I}}¼$, the sequential density is the ordinary density with respect to $\overline{\hat{I}}¼$. We also prove that if $(\hat{I}{¼}_n)$ is a sequence of invariant probability measures converging in the strong sense to an invariant probability measure $\overline{\hat{I}}¼$, then the sequential density of every rational language exists for this sequence.