Local Limit Properties for Pattern Statistics and Rational 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 rational formal series s with positive real coefficients. Our main result is a local limit theorem of Gaussian type for these statistics under the hypothesis that s is a power of a primitive series. This result is obtained by showing a general criterion for (Gaussian) local limit laws of sequences of integer random variables. To prove our result we also introduce and analyse a notion of symbol-periodicity for irreducible matrices, whose entries are polynomials over positive semirings; the properties we prove on this topic extend the classical Perron--Frobenius theory of non-negative real matrices. As a further application we obtain some asymptotic evaluations of the maximum coefficient of monomials of given size for rational series in two commutative variables.