Kybernetika 56 no. 6, 1090-1110, 2020

On typical encodings of multivariate ergodic sources

Michal KupsaDOI: 10.14736/kyb-2020-6-1090


We show that the typical coordinate-wise encoding of multivariate ergodic source into prescribed alphabets has the entropy profile close to the convolution of the entropy profile of the source and the modular polymatroid that is determined by the cardinalities of the output alphabets. We show that the proportion of the exceptional encodings that are not close to the convolution goes to zero doubly exponentially. The result holds for a class of multivariate sources that satisfy asymptotic equipartition property described via the mean fluctuation of the information functions. This class covers asymptotically mean stationary processes with ergodic mean, ergodic processes, irreducible Markov chains with an arbitrary initial distribution. We also proved that typical encodings yield the asymptotic equipartition property for the output variables. These asymptotic results are based on an explicit lower bound of the proportion of encodings that transform a multivariate random variable into a variable with the entropy profile close to the suitable convolution.


entropy, entropy rate, multivariate source, ergodic source, a.e.p. property


94A24, 94A29


  1. R. Bassoli, H. Marques, J. Rodriguez, K. W. Shum and R. Tafazolli: Network coding theory: A survey. IEEE Commun. Surveys Tutor. 15 (2013), 1950-1978.   DOI:10.1109/surv.2013.013013.00104
  2. T. M. Cover and J. A. Thomas: Elements of Information Theory. John Wiley and Sons, 2012.   DOI:10.1002/0471200611
  3. R. M. Gray and J. C. Kieffer: Asymptotically mean stationary measures. Ann. Probab. 8 (1980), 962-973.   DOI:10.1214/aop/1176994624
  4. R. M. Gray: Entropy and Information Theory. Springer Science and Business Media, 2011.   CrossRef
  5. T. Kaced: Partage de secret et théorie algorithmique de l'information. PhD. Thesis, Université Montpellier 2, 2012.   CrossRef
  6. J. C. Kieffer: A generalized Shannon-McMillan theorem for the action of an amenable group on a probability space. Ann. Probab. 3 (1975), 1031-1037.   DOI:10.1214/aop/1176996230
  7. F. Matúš: Two constructions on limits of entropy functions. IEEE Trans. Inform. Theory 53 (2007), 320-330.   DOI:10.1109/tit.2006.887090
  8. F. Matúš and L. Csirmaz: Entropy region and convolution. IEEE Trans. Inform. Theory 62 (2016), 6007-6018.   DOI:10.1109/tit.2016.2601598
  9. F. Matúš and M. Kupsa: On colorings of bivariate random sequences. In: Proc. IEEE International Symposium on Information Theory 2010, pp. 1272-1275.   DOI:10.1109/isit.2010.5513700
  10. R. W. Yeung: Information Theory and Network Coding. Springer Science and Business Media, 2008.   DOI:10.1007/978-0-387-79234-7\_1