Sequential densities of rational languages

TL;DR

This paper introduces the concept of sequential density for rational languages relative to sequences of probability measures, establishing convergence results for Bernoulli and invariant measures.

Contribution

It formalizes the notion of sequential density and proves convergence properties for sequences of Bernoulli and invariant measures.

Findings

Sequential density equals ordinary density for Bernoulli measures converging to a positive Bernoulli measure.

Sequential density exists for rational languages under converging sequences of invariant measures.

Results connect rational language densities with measure convergence in probabilistic settings.

Abstract

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