On the number of occurrences of a symbol in words of regular languages

We study the random variable Y-n representing the number of occurrences of a symbol a in a word of length n chosen at random in a regular language L subset of or equal to {a, b}*, where the random choice is defined via a non-negative rational formal series r of support L. Assuming that the transition matrix associated with r is primitive we obtain asymptotic estimates for the mean value and the variance of Y-n and present a central limit theorem for its distribution. Under a further condition on such a matrix, we also derive an asymptotic approximation of the discrete Fourier transform of Y-n that allows to prove a local limit theorem for Y-n. Further consequences of our analysis concern the growth of the coefficients in rational formal series; in particular, it turns out that, for a wide class of regular languages L, the maximum number of words of length n in L having the same number of occurrences of a given symbol is of the order of growth lambda(n)/rootn, for some constant lambda > 1.