Journal of Statistical Physics

, Volume 145, Issue 5, pp 1202–1223 | Cite as

Asymptotic Synchronization for Finite-State Sources

  • Nicholas F. Travers
  • James P. Crutchfield


We extend a recent synchronization analysis of exact finite-state sources to nonexact sources for which synchronization occurs only asymptotically. Although the proof methods are quite different, the primary results remain the same. We find that an observer’s average uncertainty in the source state vanishes exponentially fast and, as a consequence, an observer’s average uncertainty in predicting future output converges exponentially fast to the source entropy rate.


Entropy rate convergence Synchronization State estimation State uncertainty 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Travers, N., Crutchfield, J.P.: Exact synchronization for finite-state sources. J. Stat. Phys. doi: 10.1007/s10955-011-0342-4. [nlin.CD]
  2. 2.
    Crutchfield, J.P., Feldman, D.P.: Regularities unseen, randomness observed: Levels of entropy convergence. Chaos 13(1), 25–54 (2003) CrossRefMATHADSMathSciNetGoogle Scholar
  3. 3.
    Ellison, C.J., Mahoney, J.R., Crutchfield, J.P.: Prediction, retrodiction, and the amount of information stored in the present. J. Stat. Phys. 136(6), 1005–1034 (2009) CrossRefMATHADSMathSciNetGoogle Scholar
  4. 4.
    Mahoney, J.R., Ellison, C.J., Crutchfield, J.P.: Information accessibility and cryptic processes. J. Phys. A, Math. Theor. 42, 362002 (2009) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Crutchfield, J.P., Young, K.: Inferring statistical complexity. Phys. Rev. Lett. 63, 105–108 (1989) CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory, 2nd edn. Wiley-Interscience, New York (2006) MATHGoogle Scholar
  7. 7.
    Shannon, C.E., Weaver, W.: The Mathematical Theory of Communication. University of Illinois Press, Champaign-Urbana (1962) Google Scholar
  8. 8.
    Massey, J.L.: Markov information sources. In: Skwirzynski, J.K. (ed.) New Directions in Signal Processing in Communication and Control. NATO Advanced Study Institutes Series, vol. E25, pp. 15–26. Noordhoff, Leyden (1975) Google Scholar
  9. 9.
    Glynn, P.W., Ormoneit, D.: Hoeffding’s inequality for uniformly ergodic Markov chains. Stat. Probab. Lett. 56(2), 143–146 (2002) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Nicholas F. Travers
    • 1
    • 2
  • James P. Crutchfield
    • 1
    • 2
    • 3
    • 4
  1. 1.Complexity Sciences CenterUniversity of California at DavisDavisUSA
  2. 2.Mathematics DepartmentUniversity of California at DavisDavisUSA
  3. 3.Physics DepartmentUniversity of California at DavisDavisUSA
  4. 4.Santa Fe InstituteSanta FeUSA

Personalised recommendations