On the Feed-Forward Rate-Distortion Function for Stationary and Ergodic Sources

In this work we show that for a stationary and ergodic source
$X$
, stationary and ergodic test channels achieve the rate-distortion function for the lossy source coding problem with feed-forward. As a by-product, we prove that the upper bound for the feed-forward rate-distortion function derived by Venkataramanan and Pradhan (2007) is tight. In addition, we derive an upper bound on the rate-distortion function by proving a Shannon-McMillan-Breiman theorem for the causally conditional entropy rate for the class of asymptotic mean stationary and ergodic processes.