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Worst sources and robust codes for difference distortion measures

It has long been known that for a mean-square error distortion measure the Gaussian distribution requires the largest rate of all sources of a given variance. It has also been stated that a code designed for the Gaussian source and yielding distortion d when used with a Gaussian source will yield distortion \leq d when used with any independent-letter source of the same variance. In this paper, we extend these results in two directions: a) instead of assuming that the source has a fixed variance, we fix an arbitrary moment; b) instead of mean-square error distortion measures, we consider nearly arbitrary continuous difference distortion measures. For each moment constraint, we show that there is a given distribution that has the largest rate for (nearly) any difference distortion measure and that a code designed for this source yielding distortion d yields distortion \leq d for any ergodic source satisfying the same moment constraint. Furthermore, digital encoding of the output of this encoder may yield a lower rate when this encoder is used with a source for which it was not designed. We also extend these results to the case of a random process or random field of known correlation function under a difference distortion measure.