The Annals of Probability

Bounding $\bar{d}$-distance by informational divergence: a method to prove measure concentration

K. Marton

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There is a simple inequality by Pinsker between variational distance and informational divergence of probability measures defined on arbitrary probability spaces. We shall consider probability measures on sequences taken from countable alphabets, and derive, from Pinsker's inequality, bounds on the $\bar{d}$-distance by informational divergence. Such bounds can be used to prove the "concentration of measure" phenomenon for some nonproduct distributions.

Article information

Ann. Probab. Volume 24, Number 2 (1996), 857-866.

First available in Project Euclid: 11 December 2002

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Zentralblatt MATH identifier

Primary: 60F10: Large deviations 60G70: Extreme value theory; extremal processes 60G05: Foundations of stochastic processes

Measure concentration isoperimetric inequality Markov chains $\bar{d}$-distance informational divergence


Marton, K. Bounding $\bar{d}$-distance by informational divergence: a method to prove measure concentration. Ann. Probab. 24 (1996), no. 2, 857--866. doi:10.1214/aop/1039639365.

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  • AHLSWEDE, R., GACS, P. and KORNER, J. 1976. Bounds on conditional probabilities with applica´ ¨ tions in multi-user communication. Z. Wahrsch. Verw. Gebiete 34 157 177. Z.
  • CSISZAR, I. and KORNER, J. 1981. Information Theory: Coding Theorems for Discrete Memory´ ¨ less Sy stems. Academic Press, New York. Z.
  • MARTON, K. 1986. A simple proof of the blowing-up lemma. IEEE Trans. Inform. Theory IT-32 445 446. Z.
  • MARTON, K. 1995a. A concentration-of-measure inequality for contracting Markov chains. Geometric and Functional Analy sis. To appear. Z.
  • MARTON, K. 1995b. Processes having the blowing-up property. Unpublished manuscript. Z.
  • MARTON, K. and SHIELDS, P. C. 1994. The positive divergence and blowing-up properties. Israeli J. Math. 86 331 348. Z.
  • MCDIARMID, C. 1989. On the method of bounded differences. In Survey s in Combinatorics. Z. London Mathematical Society Lecture Notes J. Simons, ed. 141 148 188. Cambridge Univ. Press, London. Z.
  • PAPAMARCOU, A. and SHALABY, H. 1993. Error exponent for distributed detection of Markov sources. IEEE Trans. Inform. Theory 40 397 408. Z.
  • PINSKER, M. S. 1964. Information and Information Stability of Random Variables and Processes. Holden-Day, San Francisco. Z.
  • TALAGRAND, M. 1995. Concentration of measure and isoperimetric inequalities in product spaces. Publ. IHES. 81 73 205.