Probabilistic models for pattern statistics
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 40 (2006) no. 2, pp. 207-225.

In this work we study some probabilistic models for the random generation of words over a given alphabet used in the literature in connection with pattern statistics. Our goal is to compare models based on markovian processes (where the occurrence of a symbol in a given position only depends on a finite number of previous occurrences) and the stochastic models that can generate a word of given length from a regular language under uniform distribution. We present some results that show the differences between these two stochastic models and their relationship with the rational probabilistic measures.

DOI : 10.1051/ita:2006003
Classification : 68Q45, 68Q10, 60J99
Mots-clés : pattern statistics, Markov chains, probabilistic automata, rational formal series 
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Goldwurm, Massimiliano; Radicioni, Roberto. Probabilistic models for pattern statistics. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 40 (2006) no. 2, pp. 207-225. doi : 10.1051/ita:2006003. http://www.numdam.org/articles/10.1051/ita:2006003/

[1] I.N. Bernstein, Modules over a ring of differential operators, study of the fundamental solutions of equations with constant coefficients. Functional Anal. Appl. 5 (1971) 1-16 (Russian); pages 89-101 (English). | Zbl

[2] J. Berstel and C. Reutenauer, Rational Series and their Languages. Springer-Verlag, New York, Heidelberg, Berlin (1988). | MR | Zbl

[3] A. Bertoni, C. Choffrut, M. Goldwurm and V. Lonati, On the number of occurrences of a symbol in words of regular languages. Theoret. Comput. Sci. 302 (2003) 431-456. | Zbl

[4] A. Bertoni, C. Choffrut, M. Goldwurm and V. Lonati, Local limit properties for pattern statistics and rational models. Theory. Comput. Syst. 39 (2006) 209-235. | Zbl

[5] J. Bourdon and B. Vallée, Generalized pattern matching statistics. Mathematics and computer science II: algorithms, trees, combinatorics and probabilities, in Proc. of Versailles Colloquium. Birkhauser (2002) 249-265 | Zbl

[6] A. Denise, Génération aléatoire et uniforme de mots de langages rationnels. Theoret. Comput. Sci. 159 (1996) 43-63. | Zbl

[7] P. Flajolet, P. Zimmerman and B. Van Cutsem, A calculus for the random generation of labelled combinatorial structures. Theoret. Comput. Sci. 132 (1994) 1-35. | Zbl

[8] M. Goldwurm and V. Lonati, Pattern occurrences in multicomponent models, in Proc. 22nd STACS. Lect. Notes Comput. Sci. 3404 (2005) 680-692. | Zbl

[9] G. Hansel and D. Perrin, Rational probability measures. Theoret. Comput. Sci. 65 (1989) 171-188 (french version in Mots, edited by M. Lothaire, Hermes (1990) 335-357). | Zbl

[10] M. Iosifescu, Finite Markov Processes and Their Applications. J. Wiley and Son (1980). | MR | Zbl

[11] P. Jacket and W. Szpankowski, Analytic approach to pattern matching, Chap. 7 in Applied Combinatorics on Words. Cambridge University Press (2005).

[12] J.G. Kemeny and J.L. Snell, Finite Markov Chains. Van Nostrand (1960). | MR | Zbl

[13] L. Lipshitz, D-finite power series. J. Algebra 122 (1989) 353-373. | Zbl

[14] P. Nicodème, B. Salvy and P. Flajolet, Motif statistics. Theoret. Comput. Sci. 287 (2002) 593-617. | Zbl

[15] A. Paz, Introduction to Probabilistic Automata. Academic Press (1971). | MR | Zbl

[16] M. Régnier and W. Szpankowski, On pattern frequency occurrences in a Markovian sequence. Algorithmica 22 (1998) 621-649. | Zbl

[17] A. Salomaa and M. Soittola, Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag (1978). | MR | Zbl

[18] E. Seneta, Non-negative Matrices and Markov Chains. Springer-Verlag (1981). | MR | Zbl

[19] R.P. Stanley, Differentiably finite power series. Eur. J. Combin. 1 (1980) 175-188. | Zbl

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