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Quenched large deviation principle for words in a letter sequence
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  • Published: 17 June 2009

Quenched large deviation principle for words in a letter sequence

  • Matthias Birkner1,
  • Andreas Greven2 &
  • Frank den Hollander3,4 

Probability Theory and Related Fields volume 148, pages 403–456 (2010)Cite this article

A Correction to this article was published on 03 July 2023

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Abstract

When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In the present paper we consider the quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an algebraic tail. The rate function turns out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential tail, in which case the rate function turns out to be the first term on the set where the second term vanishes and to be infinite elsewhere. In a companion paper the annealed and the quenched LDP are applied to the collision local time of transient random walks, and the existence of an intermediate phase for a class of interacting stochastic systems is established.

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  • 03 July 2023

    A Correction to this paper has been published: https://doi.org/10.1007/s00440-023-01212-w

References

  1. Birkner, M.: Particle Systems with Locally Dependent Branching: Long-Time Behaviour, Genealogy and Critical Parameters. Dissertation, Johann Wolfgang Goethe-Universität Frankfurt am Main. http://publikationen.ub.uni-frankfurt.de/volltexte/2003/314/ (2003)

  2. Birkner M.: Conditional large deviations for a sequence of words. Stoch. Proc. Appl. 118, 703–729 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Birkner, M., Greven, A., den Hollander, F.: Large deviations for the collision local time of transient random walks and intermediate phases in interacting stochastic systems, preprint 2008

  4. Comets F.: Large deviation estimates for a conditional probability distribution. Applications to random interaction Gibbs measures. Probab. Theory Relat. Fields 80, 407–432 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dembo A., Zeitouni O.: Large Deviations Techniques and Applications, 2nd edn. Springer, Heidelberg (1998)

    MATH  Google Scholar 

  6. den Hollander F.: Large Deviations, Fields Institute Monographs 14. American Mathematical Society, Providence (2000)

    Google Scholar 

  7. Deuschel J.-D., Stroock D.W.: Large Deviations. Academic, Boston (1989)

    MATH  Google Scholar 

  8. Georgii H.-O.: Gibbs Measures and Phase Transitions, de Gruyter Studies in Mathematics 9. Walter de Gruyter, Berlin (1988)

    Google Scholar 

  9. Holley R., Liggett T.M.: Generalized potlatch and smoothing processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 55, 165–195 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  10. Shields P.C.: The Ergodic Theory of Discrete Sample Paths. American Mathematical Society, Providence, RI (1996)

    MATH  Google Scholar 

  11. Strassen V.: The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423–439 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  12. Toninelli F.: Disordered pinning models and copolymers: beyond annealed bounds. Ann. Appl. Probab. 18, 1569–1587 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Walters P.: An Introduction to Ergodic Theory, Graduate Texts in Mathematics 79. Springer, New York (1982)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department Biologie II, Abteilung Evolutionsbiologie, University of Munich (LMU), Grosshaderner Str. 2, 82152, Planegg-Martinsried, Germany

    Matthias Birkner

  2. Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstrasse 1 1/2, 91054, Erlangen, Germany

    Andreas Greven

  3. Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA, Leiden, The Netherlands

    Frank den Hollander

  4. EURANDOM, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Frank den Hollander

Authors
  1. Matthias Birkner
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  2. Andreas Greven
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  3. Frank den Hollander
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Corresponding author

Correspondence to Matthias Birkner.

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Cite this article

Birkner, M., Greven, A. & den Hollander, F. Quenched large deviation principle for words in a letter sequence. Probab. Theory Relat. Fields 148, 403–456 (2010). https://doi.org/10.1007/s00440-009-0235-5

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  • Received: 07 July 2008

  • Revised: 14 May 2009

  • Published: 17 June 2009

  • Issue Date: November 2010

  • DOI: https://doi.org/10.1007/s00440-009-0235-5

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Keywords

  • Letters and words
  • Renewal process
  • Empirical process
  • Annealed vs. quenched
  • Large deviation principle
  • Rate function
  • Specific relative entropy

Mathematics Subject Classification (2000)

  • 60F10
  • 60G10
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