Local limit properties for pattern statistics and rational models
Résumé
Motivated by problems of pattern statistics, we study the limit distribu- tion 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 eval- uations of the maximum coefficient of monomials of given size for rational series in two commutative variables.