Frequency of symbol occurrences in simple non-primitive stochastic models. Rapporto Interno n. 287-03, Dipartimento di Scienze dell’Informazione, Università degli Studi di Milano, February 2003 (available at http://homes.dsi.unimi.it/∼goldwurm/home.html

Abstract. We study the random variable Yn representing the number of occurrences of a given symbol in a word of length n generated at random. The stochastic model we assume is a simple non-ergodic model defined by the product of two primitive rational formal series, which form two distinct ergodic components. We obtain asymptotic evaluations for the mean and the variance of Yn and its limit distribution. It turns out that there are two main cases: if one component is dominant and non-degenerate we get a Gaussian limit distribution; if the two components are equipotent and have different leading terms of the mean, we get a uni-form limit distribution. Other particular limit distributions are obtained in the case of a degenerate dominant component and in the equipotent case when the leading terms of the expectation values are equal.