Pattern statistics in bicomponent stochastic models

We give asymptotic estimates of the frequency of occurrences of a symbol in a random word generated by any (non-ergodic) bicomponent stochastic model. More precisely, we consider the random variable Yn representing the number of occurrences of a given symbol in a word of length n generated at random; the stochastic model is defined by a rational formal series r having a linear representation with two primitive components. This model includes the case when r is the product or the sum of two primitive rational formal series. We obtain asymptotic evaluations for the mean and the variance of Yn and its limit distribution. These results improve the analysis presented in a recent work dealing with the particular case where r is the product of two primitive rational formal series