Local limit laws for symbol statistics in bicomponent rational models

TL;DR

This paper analyzes the local limit laws for symbol occurrence counts in words generated by rational stochastic models with two components, establishing convergence rates and conditions affecting the distribution.

Contribution

It provides a detailed analysis of local limit distributions in bicomponent rational models, including convergence rates and the influence of component communication.

Findings

Convergence rate of O(n^{-1/2}) for local limit laws.

Dependence of distributions on component parameters and communication.

Gaussian approximation for single-component models.

Abstract

We study the local limit distribution of the number of occurrences of a symbol in words of length $n$ generated at random in a regular language according to a rational stochastic model. We present an analysis of the main local limits when the finite state automaton defining the stochastic model consists of two primitive components. The limit distributions depend on several parameters and conditions, such as the main constants of mean value and variance of our statistics associated with the two components, and the existence of communications from the first to the second component. The convergence rate of these results is always of order $O(n^{-1/2})$. We also prove an analogous $O(n^{-1/2})$ convergence rate to a Gaussian density of the same statistic whenever the stochastic models only consists of one (primitive) component.