Abstract
Motivated by problems of pattern statistics, we study the limit distribution 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 evaluations of the maximum coefficient of monomials of given size for rational series in two commutative variables.
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Bertoni, A., Choffrut, C., Goldwurm, M. et al. Local Limit Properties for Pattern Statistics and Rational Models. Theory Comput Syst 39, 209–235 (2006). https://doi.org/10.1007/s00224-005-1265-8
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DOI: https://doi.org/10.1007/s00224-005-1265-8