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February, 1978 The Distortion-Rate Function for Nonergodic Sources
P. C. Shields, D. L. Neuhoff, L. D. Davisson, F. Ledrappier
Ann. Probab. 6(1): 138-143 (February, 1978). DOI: 10.1214/aop/1176995618


The distortion rate function $D(R)$ is defined as an infimum of distortion with respect to a mutual information constraint. The usual coding theorems assert that, for ergodic souces, $D(R)$ is equal to $\delta(R)$, the least distortion attainable by block codes of rate $R$. If a source has ergodic components $\{\theta\}$ with weighting measure $dw(\theta)$, it has been shown by Gray and Davisson that $\delta(R)$ is the integral of the components $\delta_\theta(R)$ with respect to $dw(\theta)$. We show that $D(R)$ is the infimum of the integrals of $D_\theta(R_\theta)$ where the integral of $R_\theta$ is $R$. Our method of proof also gives a formula for the $\bar{d}$-distance in terms of ergodic components and shows that $D(R) = D'(R)$, which is defined as the infimum of distortion subject to an entropy constraint.


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P. C. Shields. D. L. Neuhoff. L. D. Davisson. F. Ledrappier. "The Distortion-Rate Function for Nonergodic Sources." Ann. Probab. 6 (1) 138 - 143, February, 1978.


Published: February, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0375.94009
MathSciNet: MR469523
Digital Object Identifier: 10.1214/aop/1176995618

Primary: 94A20
Secondary: 28A70

Keywords: $\bar d$-distance , Distortion rate function , Entropy

Rights: Copyright © 1978 Institute of Mathematical Statistics

Vol.6 • No. 1 • February, 1978
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