Local limit laws for symbol statistics in bicomponent rational models

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 $O(n^-1/2)$. For the same statistics we also prove an analogous $O(n^-1/2)$ convergence rate of the Gaussian local limit law whenever the stochastic model consists of one primitive component.