Probability Theory and Related Fields

, Volume 104, Issue 1, pp 1–14

Refinements of the Gibbs conditioning principle

  • A. Dembo
  • O. Zeitouni
Article

Summary

Refinements of Sanov's large deviations theorem lead via Csiszár's information theoretic identity to refinements of the Gibbs conditioning principle which are valid for blocks whose length increase with the length of the conditioning sequence. Sharp bounds on the growth of the block length with the length of the conditioning sequence are derived.

Mathematics Subject Classification (1991)

60F10 

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References

  1. 1.
    Bhattacharya, R.N., Ranga Rao, R.: Normal approximation and asymptotic expansions. New York: Wiley 1976Google Scholar
  2. 2.
    Bolthausen, E.: Laplace approximations for sums of independent random vectors, Part II. Probab. Theory Related Fields76, 167–206 (1987)Google Scholar
  3. 3.
    Bolthausen, E.: Maximum entropy principles for Markov processes. In: Stochastic Processes and Applications in Mathematical Physics, Proceeding of the Bielefeld Conference, 1985. Math. Appl.61, Dordrecht: Kluwer 1990Google Scholar
  4. 4.
    Csiszár, I.:I-divergence geometry of probability distributions and minimization problems. Ann. Probab.3, 146–158 (1975)Google Scholar
  5. 5.
    Csiszár, I.: Sanov property, generalizedI-projection and a conditional limit theorem. Ann. Probab.12, 768–793 (1984)Google Scholar
  6. 6.
    Csiszár, I., Cover, T.M., Choi, B.S.: Conditional limit theorems under Markov conditioning. IEEE Trans. Inform. Theory33, 788–801 (1987)Google Scholar
  7. 7.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Boston: Jones and Bartlett 1993Google Scholar
  8. 8.
    Diaconis, P., Freedman, D.: Finite exchangeable sequences. Ann. Probab.4, 745–764 (1980)Google Scholar
  9. 9.
    Diaconis, P., Freedman, D.: Conditional limit theorems for exponential families and finite versions of de Finetti's theorem. J. Theoret. Probab.1, 381–410 (1988)Google Scholar
  10. 10.
    Dinwoodie, I.H.: Measures dominantes et théorème de Sanov. Ann. Inst. Henry Poincaré28, 365–373 (1992)Google Scholar
  11. 11.
    Iltis, M.: Asymptotics of Large Deviations inR d. Preprint, 1993Google Scholar
  12. 12.
    Korolyuk, V.S., Borovskikh, Yu.V.: Convergence rate for degenerate Von Mises functionals. Theory Probab. Appl.33, 125–135 (1988)Google Scholar
  13. 13.
    Lanford, O.E.: Entropy and equilibrium states in classical statistical mechanics. Lecture notes in Physics Vol. 20, 1–113 (1973) New York: SpringerGoogle Scholar
  14. 14.
    Ney, P.: Dominating points and the asymptotics of large deviations for random walk onR d. Ann. Probab.11, 158–167 (1983)Google Scholar
  15. 15.
    Schroeder, C.:I-projection and conditional limit theorems for discrete parameter Markov processes. Ann. Probab.21, 721–758 (1993)Google Scholar
  16. 16.
    Stroock, D.W., Zeitouni, O.: Microcanonical distributions, Gibbs states, and the equivalence of ensembles. In: Durrett, R., Kesten, H. (eds.) Festchrift in honor of F. Spitzer, pp. 399–424. Basel, Switzerland: Birkhäuser, 1991Google Scholar
  17. 17.
    von Bahr, B., Esseen, C.G.: Inequalities for therth absolute moment for a sum of random variables, 1≦r≦2. Ann. Math. Statistics36, 299–303 (1965)Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • A. Dembo
    • 1
  • O. Zeitouni
    • 2
  1. 1.Department of Mathematics and Department of StatisticsStanford UniversityStanfordUSA
  2. 2.Department of Electrical EngineeringTechnion-Israel Institute of TechnologyHaifaIsrael

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