Large deviation properties for pattern statistics in primitive rational models

We present a large deviation property for pattern statistics representing the number of occurrences of a symbol in words of given length generated at random according to a rational stochastic model. This result is proved assuming that the transition matrix of the model is primitive. We show how the rate function of the large deviation property depends on the main eigenvalues and eigenvectors of the transition matrices associated with the different symbols of the alphabet. We also yield general conditions to guarantee that the range of validity of the large deviation estimate coincides with the whole interval $(0,1)$, which represents in our context the largest possible open interval where the property may hold. The case of smaller intervals of validity is finally examined by means of examples.