Average value and variance of pattern statistics in rational models

We study the pattern statistics representing the number of occurrences of a given string in a word of length n generated at random by rational stochastic models, defined by means of weighted finite automata. We get asymptotic estimations for the mean value and the variance of these statistics under the hypothesis that the matrix of all transition weights is primitive. Our results extend previous evaluations obtained by assuming ergodic stationary Markovian sources and they yield a general framework to determine analogous estimations under several stochastic models. In particular they show the role of the stationarity hypothesis in such models.