Article

Frequency of symbol occurrences in simple non-primitive stochastic models

Authors:

Diego de Falco,

Massimiliano Goldwurm,

Violetta LonatiAuthors Info & Claims

DLT'03: Proceedings of the 7th international conference on Developments in language theory

Pages 242 - 253

Published: 07 July 2003 Publication History

Abstract

We study the random variable Yn representing the number of occurrences of a given symbol in a word of length n generated at random. The stochastic model we assume is a simple non-ergodic model defined by the product of two primitive rational formal series, which form two distinct ergodic components. We obtain asymptotic evaluations for the mean and the variance of Y_n and its limit distribution. It turns out that there are two main cases: if one component is dominant and nondegenerate we get a Gaussian limit distribution; if the two components are equipotent and have different leading terms of the mean, we get a uniform limit distribution. Other particular limit distributions are obtained in the case of a degenerate dominant component and in the equipotent case when the leading terms of the expectation values are equal.

References

[1]

E. A. Bender and F. Kochman. The distribution of subword counts is usually normal. European Journal of Combinatorics, 14:265-275, 1993.

Digital Library

[2]

J. Berstel and C. Reutenauer. Rational series and their languages, Springer-Verlag, New York - Heidelberg - Berlin, 1988.

Digital Library

[3]

A. Bertoni, C. Choffrut, M. Goldwurm, and V. Lonati. On the number of occurrences of a symbol in words of regular languages. Rapporto Interno n. 274-02, Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, February 2002 (to appear in TCS).

Digital Library

[4]

A. Bertoni, C. Choffrut, M. Goldwurm, and V. Lonati. The symbol-periodicity of irreducible finite automata. Rapporto Interno n. 277-02, Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, April 2002 (available at http://homes.dsi.unimi.it/~goldwurm/home.html).

[5]

D. de Falco, M. Goldwurm, and V. Lonati. Frequency of symbol occurrences in simple non-primitive stochastic models. Rapporto Interno n. 287-03, Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, February 2003 (available at http://homes.dsi.unimi.it/~goldwurm/home.html).

[6]

A. Denise. Génération aléatoire uniforme de mots de langages rationnels. Theoretical Computer Science, 159:43-63, 1996.

Digital Library

[7]

J. Fickett. Recognition of protein coding regions in DNA sequences. Nucleic Acid Res, 10:5303-5318, 1982.

[8]

P. Flajolet and R. Sedgewick. The average case analysis of algorithms: multivariate asymptotics and limit distributions. Rapport de recherche n. 3162, INRIA Rocquencourt, May 1997.

[9]

M.S. Gelfand. Prediction of function in DNA sequence analysis. J. Comput. Biol., 2:87-117, 1995.

[10]

L.J. Guibas and A. M. Odlyzko. Maximal prefix-synchronized codes. SIAM J. Appl. Math., 35:401-418, 1978.

[11]

L.J. Guibas and A. M. Odlyzko. Periods in strings. Journal of Combinatorial Theory. Series A, 30:19-43, 1981.

[12]

L.J. Guibas and A. M. Odlyzko. String overlaps, pattern matching, and nontransitive games. Journal of Combinatorial Theory. Series A, 30(2):183-208, 1981.

[13]

P. Jokinen and E. Ukkonen. Two algorithms for approximate string matching in static texts Proc. MFCS 91, Lecture Notes in Computer Science, vol. n.520, Springer, 240-248, 1991.

[14]

P. Nicodeme, B. Salvy, and P. Flajolet. Motif statistics. In Proceedings of the 7th ESA, J. Nešetril editor. Lecture Notes in Computer Science, vol. n.1643, Springer, 1999, 194-211.

Digital Library

[15]

B. Prum, F. Rudolphe and E. Turckheim. Finding words with unexpected frequencies in deoxyribonucleic acid sequence. J. Roy. Statist. Soc. Ser. B, 57: 205-220, 1995.

[16]

M. Régnier and W. Szpankowski. On the approximate pattern occurrence in a text. Proc. Sequence '97, Positano, 1997.

Digital Library

[17]

M. Régnier and W. Szpankowski. On pattern frequency occurrences in a Markovian sequence. Algorithmica, 22 (4):621-649, 1998.

[18]

C. Reutenauer. Propriétés arithmétiques et topologiques de séries rationnelles en variables non commutatives, These Sc. Maths, Doctorat troisieme cycle, Université Paris VI, 1977.

[19]

E. Seneta. Non-negative matrices and Markov chains, Springer-Verlag, New York Heidelberg Berlin, 1981.

[20]

M. Waterman. Introduction to computational biology, Chapman & Hall, New York, 1995.

[21]

K. Wich. Sublinear ambiguity. In Proceedings of the 25th MFCS, M. Nielsen and B. Rovan editors. Lecture Notes in Computer Science, vol. n.1893, Springer, 2000, 690-698.

Digital Library

[22]

S. Wolfram. The Mathematica book Fourth Edition, Wolfram Media - Cambridge University Press, 1999.

Digital Library

Cited By

Choffrut CGoldwurm MLonati V(2004)On the maximum coefficients of rational formal series in commuting variablesProceedings of the 8th international conference on Developments in Language Theory10.1007/978-3-540-30550-7_10(114-126)Online publication date: 13-Dec-2004
https://dl.acm.org/doi/10.1007/978-3-540-30550-7_10

Recommendations

Frequency of symbol occurrences in bicomponent stochastic models
Developments in language theory

We give asymptotic estimates of the frequency of occurrences of a symbol in a random word generated by any bicomponent stochastic model. More precisely, we consider the random variable Y_n, representing the number of occurrences of a given symbol in a ...
Stationarity in models of successive stochastic phenomena

It is well known that the image of a stationary stochastic process under a deterministic, measurable function that ignores the past is again a stationary stochastic process. This result allows us to assert the stationarity of a model for a phenomenon ...
Pattern occurrences in multicomponent models
STACS'05: Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science

In this paper we determine some limit distributions of pattern statistics in rational stochastic models, defined by means of nondeterministic weighted finite automata. We present a general approach to analyse these statistics in rational models having ...

Comments

Information & Contributors

Information

Published In

cover image Guide Proceedings

DLT'03: Proceedings of the 7th international conference on Developments in language theory

July 2003

437 pages

ISBN:3540404341

Editors:
Zoltán Ésik
University of Szeged, Department of Informatics, Szeged, Hungary
,
Zoltán Fülöp
University of Szeged, Department of Informatics, Szeged, Hungary

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 07 July 2003

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

1
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 24 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Choffrut CGoldwurm MLonati V(2004)On the maximum coefficients of rational formal series in commuting variablesProceedings of the 8th international conference on Developments in Language Theory10.1007/978-3-540-30550-7_10(114-126)Online publication date: 13-Dec-2004
https://dl.acm.org/doi/10.1007/978-3-540-30550-7_10

View Options

View options

Media

Figures

Other

Tables

View Table of Contents