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# The Distortion-Rate Function for Nonergodic Sources

P. C. Shields, D. L. Neuhoff, L. D. Davisson and F. Ledrappier
The Annals of Probability
Vol. 6, No. 1 (Feb., 1978), pp. 138-143
Stable URL: http://www.jstor.org/stable/2242868
Page Count: 6

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## Abstract

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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