A Local Limit Property for Pattern Statistics in Bicomponent Stochastic Models

We present a non-Gaussian local limit theorem for the number of occurrences of a given symbol in a word of length n generated at random. The stochastic model for the random generation is defined by a rational formal series with non-negative real coefficients. The result yields a local limit towards a uniform density function and holds under the assumption that the formal series defining the model is recognized by a weighted finite state automaton with two primitive components having equal dominant eigenvalue.