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# A Generalization of Ornstein's $\bar d$ Distance with Applications to Information Theory

Robert M. Gray, David L. Neuhoff and Paul C. Shields
The Annals of Probability
Vol. 3, No. 2 (Apr., 1975), pp. 315-328
Stable URL: http://www.jstor.org/stable/2959395
Page Count: 14
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## Abstract

Ornstein's $\bar{d}$ distance between finite alphabet discrete-time random processes is generalized in a natural way to discrete-time random processes having separable metric spaces for alphabets. As an application, several new results are obtained on the information theoretic problem of source coding with a fidelity criterion (information transmission at rates below capacity) when the source statistics are inaccurately or incompletely known. Two examples of evaluation and bounding of the process distance are presented: (i) the $\bar{d}$ distance between two binary Bernoulli shifts, and (ii) the process distance between two stationary Gaussian time series with an alphabet metric $|x - y|$.

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