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

  • Authors:

  • Alberto Bertoni;Christian Choffrut;Massimiliano Goldwurm;Violetta Lonati

  • Affiliations:

  • Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, Via Comelico 39, I-20135 Milano, Italy;Laboratoire d'Informatique Algorithmique, Fondements et Applications, Université Paris VII, 2 Place Jussieu, 75251 Paris, France;Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, Via Comelico 39, I-20135 Milano, Italy;Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, Via Comelico 39, I-20135 Milano, Italy

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2003

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Abstract

We study the random variable Yn representing the number of occurrences of a symbol a in a word of length n chosen at random in a regular language L ⊆ {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 Yn 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 Yn, that allows to prove a local limit theorem for Yn. 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 λn/√n, for some constant λ1.