A Local Limit Property for Pattern Statistics in Bicomponent Stochastic Models

  • Massimiliano Goldwurm
  • Jianyi LinEmail author
  • Marco Vignati
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10952)


We present a non-Gaussian local limit theorem for the number of occurrences of a given symbol in a word of length n generated at random. The stochastic model for the random generation is defined by a rational formal series with non-negative real coefficients. The result yields a local limit towards a uniform density function and holds under the assumption that the formal series defining the model is recognized by a weighted finite state automaton with two primitive components having equal dominant eigenvalue.


  1. 1.
    Bender, E.A.: Central and local limit theorems applied to asymptotic enumeration. J. Comb. Theory 15, 91–111 (1973)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Berstel, J., Reutenauer, C.: Rational Series and Their Languages. Springer, New York (1988)CrossRefGoogle Scholar
  3. 3.
    Bertoni, A., Choffrut, C., Goldwurm, M., Lonati, V.: On the number of occurrences of a symbol in words of regular languages. Theoret. Comput. Sci. 302, 431–456 (2003)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bertoni, A., Choffrut, C., Goldwurm, M., Lonati, V.: Local limit properties for pattern statistics and rational models. Theory Comput. Syst. 39, 209–235 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Broda, S., Machiavelo, A., Moreira, N., Reis, R.: A hitchhiker’s guide to descriptional complexity through analytic combinatorics. Theory Comput. Syst. 528, 85–100 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Broda, S., Machiavelo, A., Moreira, N., Reis, R.: On the average complexity of strong star normal form. In: Pighizzini, G., Câmpeanu, C. (eds.) DCFS 2017. LNCS, vol. 10316, pp. 77–88. Springer, Cham (2017). Scholar
  7. 7.
    de Falco, D., Goldwurm, M., Lonati, V.: Frequency of symbol occurrences in bicomponent stochastic models. Theoret. Comput. Sci. 327(3), 269–300 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Denise, A.: Génération aléatoire uniforme de mots de langages rationnels. Theoret. Comput. Sci. 159, 43–63 (1996)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  10. 10.
    Gnedenko, B.V.: Theory of Probability. Gordon and Breach Science Publisher, Amsterdam (1997)zbMATHGoogle Scholar
  11. 11.
    Nicodeme, P., Salvy, B., Flajolet, P.: Motif statistics. Theoret. Comput. Sci. 287(2), 593–617 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Prum, B., Rudolphe, F., Turckheim, E.: Finding words with unexpected frequencies in deoxyribonucleic acid sequence. J. Roy. Stat. Soc. Ser. B 57, 205–220 (1995)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Régnier, M., Szpankowski, W.: On pattern frequency occurrences in a Markovian sequence. Algorithmica 22(4), 621–649 (1998)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Salomaa, A., Soittola, M.: Automata-Theoretic Aspects of Formal Power Series. Springer, New York (1978). Scholar
  15. 15.
    Seneta, E.: Non-negative Matrices and Markov Chains. Springer, New York (1981). Scholar

Copyright information

© IFIP International Federation for Information Processing 2018

Authors and Affiliations

  • Massimiliano Goldwurm
    • 1
  • Jianyi Lin
    • 2
    Email author
  • Marco Vignati
    • 1
  1. 1.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanoItaly
  2. 2.Department of MathematicsKhalifa UniversityAbu DhabiUnited Arab Emirates

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