Local limit distributions in pattern statistics: beyond the Markovian models

Motivated by problems of pattern statistics, we study the limit distribution of the random variable counting the number of occurrences of the symbol a in a word of length n chosen at random in {a,b}*, according to a probability distribution defined via a finite automaton equipped with positive real weights. We determine the local limit distribution of such a quantity under the hypothesis that the transition matrix naturally associated with the finite automaton is primitive. Our probabilistic model extends the Markovian models traditionally used in the literature on pattern statistics. This result is obtained by introducing a notion of symbol-periodicity for irreducible matrices whose entries are polynomials in one variable over an arbitrary positive semiring. This notion and the related results,we prove are of interest in their own right, since they extend classical properties of the Perron-Frobenius Theory for non-negative real matrices.