Stationarity in the Realizations of the Causal Rate-Distortion Function for One-Sided Stationary Sources
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This paper derives novel results on the characterization of the the causal information rate-distortion function (IRDF) R_c^it(D) for arbitrarily-distributed one-sided stationary κ-th order Markov source x(1),x(2),.... It is first shown that Gorbunov and Pinsker's results on the stationarity of the realizations to the causal IRDF (stated for two-sided stationary sources) do not apply to the commonly used family of asymptotic average single-letter (AASL) distortion criteria. Moreover, we show that, in general, a reconstruction sequence cannot be both jointly stationary with a one-sided stationary source sequence and causally related to it. This implies that, in general, the causal IRDF for one-sided stationary sources cannot be realized by a stationary distribution. However, we prove that for an arbitrarily distributed one-sided stationary source and a large class of distortion criteria (including AASL), the search for R_c^it(D) can be restricted to distributions which yield the output sequence y(1), y(2),... jointly stationary with the source after κ samples. Finally, we improve the definition of the stationary causal IRDF R_c^it(D) previously introduced by Derpich and Østergaard for two-sided Markovian stationary sources and show that R_c^it(D) for a two-sided source ...,x(-1),x(0),x(1),... equals R_c^it(D) for the associated one-sided source x(1), x(2),.... This implies that, for the Gaussian quadratic case, the practical zero-delay encoder-decoder pairs proposed by Derpich and Østergaard for approaching R_c^it(D) achieve an operational data rate which exceeds R_c^it(D) by less than 1+0.5 _2(2 π e /12) ≃ 1.254 bits per sample.
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