Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sequential densities of rational languages

Alexi Block Gorman, Dominique Perrin

Abstract

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.

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 is a sequence of Bernoulli measures converging to a positive Bernoulli measure , the sequential density is the ordinary density with respect to . We also prove that if is a sequence of invariant probability measures converging in the strong sense to an invariant probability measure , then the sequential density of every rational language exists for this sequence.
Paper Structure (11 sections, 8 theorems, 33 equations, 2 figures)

This paper contains 11 sections, 8 theorems, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Shift spaces
  4. Rational languages
  5. Probability measures
  6. Sequential densities
  7. A formula for sequential densities
  8. Convergence in the strong sense
  9. Densities in more general cases
  10. Combinatorial density
  11. Acknowledgements

Key Result

Theorem 3.1

If $\mu$ is a convergent sequence of Bernoulli measures on $A^\mathbb{Z}$, the sequential density of every rational language exists. If $L$ is aperiodic, the density exists in the strong sense.

Figures (2)

  • Figure 4.1: The automaton $\mathcal{A}$.
  • Figure 6.1: The shift $X_2$.

Theorems & Definitions (13)

  • Example 2.1
  • Theorem 3.1
  • Example 3.2
  • Example 3.3
  • Theorem 4.1
  • Example 4.2
  • Corollary 4.3
  • Example 4.4
  • Theorem 5.1
  • Theorem 5.2
  • ...and 3 more