Probability Theory and Related Fields

, Volume 148, Issue 3–4, pp 403–456 | Cite as

Quenched large deviation principle for words in a letter sequence

  • Matthias BirknerEmail author
  • Andreas Greven
  • Frank den Hollander


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.


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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  1. 1.
    Birkner, M.: Particle Systems with Locally Dependent Branching: Long-Time Behaviour, Genealogy and Critical Parameters. Dissertation, Johann Wolfgang Goethe-Universität Frankfurt am Main. (2003)
  2. 2.
    Birkner M.: Conditional large deviations for a sequence of words. Stoch. Proc. Appl. 118, 703–729 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 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 2008Google Scholar
  4. 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)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Dembo A., Zeitouni O.: Large Deviations Techniques and Applications, 2nd edn. Springer, Heidelberg (1998)zbMATHGoogle Scholar
  6. 6.
    den Hollander F.: Large Deviations, Fields Institute Monographs 14. American Mathematical Society, Providence (2000)Google Scholar
  7. 7.
    Deuschel J.-D., Stroock D.W.: Large Deviations. Academic, Boston (1989)zbMATHGoogle Scholar
  8. 8.
    Georgii H.-O.: Gibbs Measures and Phase Transitions, de Gruyter Studies in Mathematics 9. Walter de Gruyter, Berlin (1988)Google Scholar
  9. 9.
    Holley R., Liggett T.M.: Generalized potlatch and smoothing processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 55, 165–195 (1981)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Shields P.C.: The Ergodic Theory of Discrete Sample Paths. American Mathematical Society, Providence, RI (1996)zbMATHGoogle Scholar
  11. 11.
    Strassen V.: The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423–439 (1965)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Toninelli F.: Disordered pinning models and copolymers: beyond annealed bounds. Ann. Appl. Probab. 18, 1569–1587 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Walters P.: An Introduction to Ergodic Theory, Graduate Texts in Mathematics 79. Springer, New York (1982)Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Matthias Birkner
    • 1
    Email author
  • Andreas Greven
    • 2
  • Frank den Hollander
    • 3
    • 4
  1. 1.Department Biologie II, Abteilung EvolutionsbiologieUniversity of Munich (LMU)Planegg-MartinsriedGermany
  2. 2.Mathematisches InstitutUniversität Erlangen-NürnbergErlangenGermany
  3. 3.Mathematical InstituteLeiden UniversityLeidenThe Netherlands
  4. 4.EURANDOMEindhovenThe Netherlands

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