Sequential densities of rational languages
This work addresses theoretical aspects of formal language theory and probability, likely incremental for researchers in automata and measure theory.
The paper tackles the problem of defining and analyzing the density of rational languages with respect to sequences of probability measures, proving that under certain convergence conditions (e.g., Bernoulli or invariant measures), the sequential density equals or exists as the ordinary density.
We introduce the notion of density of a rational language with respect to a sequence of probability measures. We prove that if $(μ_n)$ is a sequence of Bernoulli measures converging to a positive Bernoulli measure $\overlineμ$, the sequential density is the ordinary density with respect to $\overlineμ$. We also prove that if $(μ_n)$ is a sequence of invariant probability measures converging in the strong sense to an invariant probability measure $\overlineμ$, then the sequential density of every rational language exists for this sequence.