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Article Dans Une Revue Theoretical Computer Science Année : 2003

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

Résumé

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 Lsubset 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 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 Image, for some constant λ>1.
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Dates et versions

hal-00158274 , version 1 (28-06-2007)

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Christian Choffrut, Violetta Lonati, Massimiliano Goldwurm, Alberto Bertoni. On the number of occurrences of a symbol in words of regular languages. Theoretical Computer Science, 2003, 302 (1-3), pp.431-456. ⟨10.1016/S0304-3975⟩. ⟨hal-00158274⟩
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